Properties

Label 177.8.a.a.1.15
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(-18.7189\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+18.7189 q^{2} -27.0000 q^{3} +222.398 q^{4} +443.832 q^{5} -505.411 q^{6} -1695.79 q^{7} +1767.03 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+18.7189 q^{2} -27.0000 q^{3} +222.398 q^{4} +443.832 q^{5} -505.411 q^{6} -1695.79 q^{7} +1767.03 q^{8} +729.000 q^{9} +8308.06 q^{10} -2763.59 q^{11} -6004.75 q^{12} -10956.4 q^{13} -31743.3 q^{14} -11983.5 q^{15} +4610.01 q^{16} -17168.5 q^{17} +13646.1 q^{18} -41280.8 q^{19} +98707.5 q^{20} +45786.3 q^{21} -51731.4 q^{22} +82535.2 q^{23} -47709.9 q^{24} +118862. q^{25} -205092. q^{26} -19683.0 q^{27} -377140. q^{28} +103089. q^{29} -224318. q^{30} -162172. q^{31} -139886. q^{32} +74616.9 q^{33} -321376. q^{34} -752645. q^{35} +162128. q^{36} +446923. q^{37} -772732. q^{38} +295822. q^{39} +784266. q^{40} +371488. q^{41} +857070. q^{42} -790401. q^{43} -614617. q^{44} +323554. q^{45} +1.54497e6 q^{46} -1.11011e6 q^{47} -124470. q^{48} +2.05215e6 q^{49} +2.22497e6 q^{50} +463549. q^{51} -2.43668e6 q^{52} -771292. q^{53} -368445. q^{54} -1.22657e6 q^{55} -2.99652e6 q^{56} +1.11458e6 q^{57} +1.92972e6 q^{58} +205379. q^{59} -2.66510e6 q^{60} -211813. q^{61} -3.03568e6 q^{62} -1.23623e6 q^{63} -3.20860e6 q^{64} -4.86279e6 q^{65} +1.39675e6 q^{66} +1.48116e6 q^{67} -3.81824e6 q^{68} -2.22845e6 q^{69} -1.40887e7 q^{70} +3.40336e6 q^{71} +1.28817e6 q^{72} -3.93442e6 q^{73} +8.36592e6 q^{74} -3.20927e6 q^{75} -9.18078e6 q^{76} +4.68646e6 q^{77} +5.53747e6 q^{78} +4.55385e6 q^{79} +2.04607e6 q^{80} +531441. q^{81} +6.95386e6 q^{82} -4.51953e6 q^{83} +1.01828e7 q^{84} -7.61992e6 q^{85} -1.47955e7 q^{86} -2.78342e6 q^{87} -4.88336e6 q^{88} -4.21568e6 q^{89} +6.05658e6 q^{90} +1.85797e7 q^{91} +1.83557e7 q^{92} +4.37864e6 q^{93} -2.07800e7 q^{94} -1.83217e7 q^{95} +3.77692e6 q^{96} -1.59925e6 q^{97} +3.84141e7 q^{98} -2.01466e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18.7189 1.65454 0.827268 0.561808i \(-0.189894\pi\)
0.827268 + 0.561808i \(0.189894\pi\)
\(3\) −27.0000 −0.577350
\(4\) 222.398 1.73749
\(5\) 443.832 1.58790 0.793951 0.607982i \(-0.208021\pi\)
0.793951 + 0.607982i \(0.208021\pi\)
\(6\) −505.411 −0.955246
\(7\) −1695.79 −1.86865 −0.934326 0.356419i \(-0.883997\pi\)
−0.934326 + 0.356419i \(0.883997\pi\)
\(8\) 1767.03 1.22020
\(9\) 729.000 0.333333
\(10\) 8308.06 2.62724
\(11\) −2763.59 −0.626036 −0.313018 0.949747i \(-0.601340\pi\)
−0.313018 + 0.949747i \(0.601340\pi\)
\(12\) −6004.75 −1.00314
\(13\) −10956.4 −1.38314 −0.691568 0.722311i \(-0.743080\pi\)
−0.691568 + 0.722311i \(0.743080\pi\)
\(14\) −31743.3 −3.09175
\(15\) −11983.5 −0.916775
\(16\) 4610.01 0.281373
\(17\) −17168.5 −0.847541 −0.423771 0.905770i \(-0.639294\pi\)
−0.423771 + 0.905770i \(0.639294\pi\)
\(18\) 13646.1 0.551512
\(19\) −41280.8 −1.38074 −0.690368 0.723459i \(-0.742551\pi\)
−0.690368 + 0.723459i \(0.742551\pi\)
\(20\) 98707.5 2.75896
\(21\) 45786.3 1.07887
\(22\) −51731.4 −1.03580
\(23\) 82535.2 1.41446 0.707232 0.706982i \(-0.249944\pi\)
0.707232 + 0.706982i \(0.249944\pi\)
\(24\) −47709.9 −0.704481
\(25\) 118862. 1.52143
\(26\) −205092. −2.28845
\(27\) −19683.0 −0.192450
\(28\) −377140. −3.24676
\(29\) 103089. 0.784912 0.392456 0.919771i \(-0.371625\pi\)
0.392456 + 0.919771i \(0.371625\pi\)
\(30\) −224318. −1.51684
\(31\) −162172. −0.977709 −0.488854 0.872365i \(-0.662585\pi\)
−0.488854 + 0.872365i \(0.662585\pi\)
\(32\) −139886. −0.754656
\(33\) 74616.9 0.361442
\(34\) −321376. −1.40229
\(35\) −752645. −2.96724
\(36\) 162128. 0.579162
\(37\) 446923. 1.45053 0.725265 0.688470i \(-0.241717\pi\)
0.725265 + 0.688470i \(0.241717\pi\)
\(38\) −772732. −2.28448
\(39\) 295822. 0.798554
\(40\) 784266. 1.93755
\(41\) 371488. 0.841785 0.420893 0.907111i \(-0.361717\pi\)
0.420893 + 0.907111i \(0.361717\pi\)
\(42\) 857070. 1.78502
\(43\) −790401. −1.51603 −0.758015 0.652237i \(-0.773831\pi\)
−0.758015 + 0.652237i \(0.773831\pi\)
\(44\) −614617. −1.08773
\(45\) 323554. 0.529301
\(46\) 1.54497e6 2.34028
\(47\) −1.11011e6 −1.55964 −0.779818 0.626007i \(-0.784688\pi\)
−0.779818 + 0.626007i \(0.784688\pi\)
\(48\) −124470. −0.162451
\(49\) 2.05215e6 2.49186
\(50\) 2.22497e6 2.51726
\(51\) 463549. 0.489328
\(52\) −2.43668e6 −2.40318
\(53\) −771292. −0.711628 −0.355814 0.934557i \(-0.615796\pi\)
−0.355814 + 0.934557i \(0.615796\pi\)
\(54\) −368445. −0.318415
\(55\) −1.22657e6 −0.994083
\(56\) −2.99652e6 −2.28012
\(57\) 1.11458e6 0.797168
\(58\) 1.92972e6 1.29867
\(59\) 205379. 0.130189
\(60\) −2.66510e6 −1.59288
\(61\) −211813. −0.119481 −0.0597403 0.998214i \(-0.519027\pi\)
−0.0597403 + 0.998214i \(0.519027\pi\)
\(62\) −3.03568e6 −1.61765
\(63\) −1.23623e6 −0.622884
\(64\) −3.20860e6 −1.52998
\(65\) −4.86279e6 −2.19628
\(66\) 1.39675e6 0.598018
\(67\) 1.48116e6 0.601644 0.300822 0.953680i \(-0.402739\pi\)
0.300822 + 0.953680i \(0.402739\pi\)
\(68\) −3.81824e6 −1.47259
\(69\) −2.22845e6 −0.816641
\(70\) −1.40887e7 −4.90940
\(71\) 3.40336e6 1.12851 0.564253 0.825602i \(-0.309164\pi\)
0.564253 + 0.825602i \(0.309164\pi\)
\(72\) 1.28817e6 0.406732
\(73\) −3.93442e6 −1.18372 −0.591862 0.806040i \(-0.701607\pi\)
−0.591862 + 0.806040i \(0.701607\pi\)
\(74\) 8.36592e6 2.39995
\(75\) −3.20927e6 −0.878399
\(76\) −9.18078e6 −2.39901
\(77\) 4.68646e6 1.16984
\(78\) 5.53747e6 1.32124
\(79\) 4.55385e6 1.03916 0.519582 0.854421i \(-0.326088\pi\)
0.519582 + 0.854421i \(0.326088\pi\)
\(80\) 2.04607e6 0.446793
\(81\) 531441. 0.111111
\(82\) 6.95386e6 1.39276
\(83\) −4.51953e6 −0.867602 −0.433801 0.901009i \(-0.642828\pi\)
−0.433801 + 0.901009i \(0.642828\pi\)
\(84\) 1.01828e7 1.87452
\(85\) −7.61992e6 −1.34581
\(86\) −1.47955e7 −2.50833
\(87\) −2.78342e6 −0.453169
\(88\) −4.88336e6 −0.763887
\(89\) −4.21568e6 −0.633874 −0.316937 0.948447i \(-0.602654\pi\)
−0.316937 + 0.948447i \(0.602654\pi\)
\(90\) 6.05658e6 0.875746
\(91\) 1.85797e7 2.58460
\(92\) 1.83557e7 2.45761
\(93\) 4.37864e6 0.564480
\(94\) −2.07800e7 −2.58047
\(95\) −1.83217e7 −2.19247
\(96\) 3.77692e6 0.435701
\(97\) −1.59925e6 −0.177916 −0.0889579 0.996035i \(-0.528354\pi\)
−0.0889579 + 0.996035i \(0.528354\pi\)
\(98\) 3.84141e7 4.12287
\(99\) −2.01466e6 −0.208679
\(100\) 2.64347e7 2.64347
\(101\) −1.38691e6 −0.133944 −0.0669719 0.997755i \(-0.521334\pi\)
−0.0669719 + 0.997755i \(0.521334\pi\)
\(102\) 8.67714e6 0.809611
\(103\) −1.11202e6 −0.100273 −0.0501363 0.998742i \(-0.515966\pi\)
−0.0501363 + 0.998742i \(0.515966\pi\)
\(104\) −1.93603e7 −1.68770
\(105\) 2.03214e7 1.71313
\(106\) −1.44378e7 −1.17741
\(107\) −2.40434e7 −1.89737 −0.948685 0.316223i \(-0.897585\pi\)
−0.948685 + 0.316223i \(0.897585\pi\)
\(108\) −4.37747e6 −0.334379
\(109\) 1.06281e7 0.786070 0.393035 0.919524i \(-0.371425\pi\)
0.393035 + 0.919524i \(0.371425\pi\)
\(110\) −2.29601e7 −1.64475
\(111\) −1.20669e7 −0.837463
\(112\) −7.81761e6 −0.525788
\(113\) 1.82364e7 1.18895 0.594477 0.804112i \(-0.297359\pi\)
0.594477 + 0.804112i \(0.297359\pi\)
\(114\) 2.08638e7 1.31894
\(115\) 3.66318e7 2.24603
\(116\) 2.29269e7 1.36377
\(117\) −7.98719e6 −0.461045
\(118\) 3.84447e6 0.215402
\(119\) 2.91141e7 1.58376
\(120\) −2.11752e7 −1.11865
\(121\) −1.18497e7 −0.608079
\(122\) −3.96491e6 −0.197685
\(123\) −1.00302e7 −0.486005
\(124\) −3.60667e7 −1.69876
\(125\) 1.80803e7 0.827982
\(126\) −2.31409e7 −1.03058
\(127\) 1.28606e7 0.557119 0.278559 0.960419i \(-0.410143\pi\)
0.278559 + 0.960419i \(0.410143\pi\)
\(128\) −4.21561e7 −1.77675
\(129\) 2.13408e7 0.875281
\(130\) −9.10262e7 −3.63383
\(131\) −1.03027e7 −0.400405 −0.200203 0.979754i \(-0.564160\pi\)
−0.200203 + 0.979754i \(0.564160\pi\)
\(132\) 1.65947e7 0.628000
\(133\) 7.00035e7 2.58011
\(134\) 2.77257e7 0.995442
\(135\) −8.73595e6 −0.305592
\(136\) −3.03373e7 −1.03417
\(137\) −4.73008e7 −1.57161 −0.785807 0.618471i \(-0.787752\pi\)
−0.785807 + 0.618471i \(0.787752\pi\)
\(138\) −4.17142e7 −1.35116
\(139\) 5.76637e7 1.82117 0.910585 0.413322i \(-0.135631\pi\)
0.910585 + 0.413322i \(0.135631\pi\)
\(140\) −1.67387e8 −5.15553
\(141\) 2.99729e7 0.900456
\(142\) 6.37073e7 1.86715
\(143\) 3.02789e7 0.865893
\(144\) 3.36070e6 0.0937910
\(145\) 4.57544e7 1.24636
\(146\) −7.36480e7 −1.95851
\(147\) −5.54082e7 −1.43868
\(148\) 9.93949e7 2.52028
\(149\) 9.45541e6 0.234168 0.117084 0.993122i \(-0.462645\pi\)
0.117084 + 0.993122i \(0.462645\pi\)
\(150\) −6.00741e7 −1.45334
\(151\) −1.54572e7 −0.365353 −0.182677 0.983173i \(-0.558476\pi\)
−0.182677 + 0.983173i \(0.558476\pi\)
\(152\) −7.29446e7 −1.68477
\(153\) −1.25158e7 −0.282514
\(154\) 8.77255e7 1.93555
\(155\) −7.19770e7 −1.55251
\(156\) 6.57903e7 1.38748
\(157\) 3.62627e7 0.747845 0.373922 0.927460i \(-0.378013\pi\)
0.373922 + 0.927460i \(0.378013\pi\)
\(158\) 8.52432e7 1.71933
\(159\) 2.08249e7 0.410859
\(160\) −6.20858e7 −1.19832
\(161\) −1.39962e8 −2.64314
\(162\) 9.94801e6 0.183837
\(163\) 5.27645e7 0.954301 0.477151 0.878822i \(-0.341670\pi\)
0.477151 + 0.878822i \(0.341670\pi\)
\(164\) 8.26183e7 1.46259
\(165\) 3.31174e7 0.573934
\(166\) −8.46008e7 −1.43548
\(167\) −7.36648e6 −0.122392 −0.0611959 0.998126i \(-0.519491\pi\)
−0.0611959 + 0.998126i \(0.519491\pi\)
\(168\) 8.09059e7 1.31643
\(169\) 5.72936e7 0.913066
\(170\) −1.42637e8 −2.22669
\(171\) −3.00937e7 −0.460245
\(172\) −1.75784e8 −2.63408
\(173\) 7.87542e7 1.15641 0.578206 0.815891i \(-0.303753\pi\)
0.578206 + 0.815891i \(0.303753\pi\)
\(174\) −5.21025e7 −0.749785
\(175\) −2.01564e8 −2.84303
\(176\) −1.27402e7 −0.176149
\(177\) −5.54523e6 −0.0751646
\(178\) −7.89130e7 −1.04877
\(179\) 2.13413e7 0.278121 0.139061 0.990284i \(-0.455592\pi\)
0.139061 + 0.990284i \(0.455592\pi\)
\(180\) 7.19577e7 0.919653
\(181\) 1.72243e7 0.215907 0.107953 0.994156i \(-0.465570\pi\)
0.107953 + 0.994156i \(0.465570\pi\)
\(182\) 3.47792e8 4.27631
\(183\) 5.71894e6 0.0689822
\(184\) 1.45843e8 1.72592
\(185\) 1.98359e8 2.30330
\(186\) 8.19634e7 0.933953
\(187\) 4.74466e7 0.530591
\(188\) −2.46886e8 −2.70984
\(189\) 3.33782e7 0.359622
\(190\) −3.42963e8 −3.62752
\(191\) 2.66378e7 0.276619 0.138309 0.990389i \(-0.455833\pi\)
0.138309 + 0.990389i \(0.455833\pi\)
\(192\) 8.66321e7 0.883333
\(193\) −1.69658e8 −1.69873 −0.849365 0.527807i \(-0.823015\pi\)
−0.849365 + 0.527807i \(0.823015\pi\)
\(194\) −2.99362e7 −0.294368
\(195\) 1.31295e8 1.26803
\(196\) 4.56396e8 4.32957
\(197\) 6.41433e7 0.597750 0.298875 0.954292i \(-0.403389\pi\)
0.298875 + 0.954292i \(0.403389\pi\)
\(198\) −3.77122e7 −0.345266
\(199\) −1.14787e8 −1.03254 −0.516269 0.856427i \(-0.672679\pi\)
−0.516269 + 0.856427i \(0.672679\pi\)
\(200\) 2.10033e8 1.85645
\(201\) −3.99913e7 −0.347359
\(202\) −2.59614e7 −0.221615
\(203\) −1.74818e8 −1.46673
\(204\) 1.03093e8 0.850201
\(205\) 1.64878e8 1.33667
\(206\) −2.08158e7 −0.165905
\(207\) 6.01682e7 0.471488
\(208\) −5.05090e7 −0.389177
\(209\) 1.14083e8 0.864389
\(210\) 3.80395e8 2.83444
\(211\) 6.79484e6 0.0497956 0.0248978 0.999690i \(-0.492074\pi\)
0.0248978 + 0.999690i \(0.492074\pi\)
\(212\) −1.71534e8 −1.23644
\(213\) −9.18908e7 −0.651543
\(214\) −4.50066e8 −3.13926
\(215\) −3.50805e8 −2.40731
\(216\) −3.47805e7 −0.234827
\(217\) 2.75009e8 1.82700
\(218\) 1.98946e8 1.30058
\(219\) 1.06229e8 0.683423
\(220\) −2.72787e8 −1.72721
\(221\) 1.88104e8 1.17227
\(222\) −2.25880e8 −1.38561
\(223\) 2.78892e8 1.68410 0.842052 0.539396i \(-0.181347\pi\)
0.842052 + 0.539396i \(0.181347\pi\)
\(224\) 2.37217e8 1.41019
\(225\) 8.66503e7 0.507144
\(226\) 3.41367e8 1.96717
\(227\) 4.19942e7 0.238286 0.119143 0.992877i \(-0.461985\pi\)
0.119143 + 0.992877i \(0.461985\pi\)
\(228\) 2.47881e8 1.38507
\(229\) −1.50761e7 −0.0829591 −0.0414795 0.999139i \(-0.513207\pi\)
−0.0414795 + 0.999139i \(0.513207\pi\)
\(230\) 6.85707e8 3.71613
\(231\) −1.26534e8 −0.675409
\(232\) 1.82163e8 0.957748
\(233\) 5.24867e7 0.271834 0.135917 0.990720i \(-0.456602\pi\)
0.135917 + 0.990720i \(0.456602\pi\)
\(234\) −1.49512e8 −0.762816
\(235\) −4.92702e8 −2.47655
\(236\) 4.56759e7 0.226201
\(237\) −1.22954e8 −0.599962
\(238\) 5.44985e8 2.62039
\(239\) 1.82265e8 0.863597 0.431799 0.901970i \(-0.357879\pi\)
0.431799 + 0.901970i \(0.357879\pi\)
\(240\) −5.52439e7 −0.257956
\(241\) −5.51242e7 −0.253678 −0.126839 0.991923i \(-0.540483\pi\)
−0.126839 + 0.991923i \(0.540483\pi\)
\(242\) −2.21815e8 −1.00609
\(243\) −1.43489e7 −0.0641500
\(244\) −4.71068e7 −0.207596
\(245\) 9.10812e8 3.95683
\(246\) −1.87754e8 −0.804112
\(247\) 4.52288e8 1.90975
\(248\) −2.86563e8 −1.19300
\(249\) 1.22027e8 0.500910
\(250\) 3.38444e8 1.36993
\(251\) −3.50809e8 −1.40027 −0.700137 0.714009i \(-0.746878\pi\)
−0.700137 + 0.714009i \(0.746878\pi\)
\(252\) −2.74935e8 −1.08225
\(253\) −2.28093e8 −0.885505
\(254\) 2.40736e8 0.921772
\(255\) 2.05738e8 0.777005
\(256\) −3.78416e8 −1.40971
\(257\) −3.34359e8 −1.22871 −0.614353 0.789032i \(-0.710583\pi\)
−0.614353 + 0.789032i \(0.710583\pi\)
\(258\) 3.99477e8 1.44818
\(259\) −7.57886e8 −2.71053
\(260\) −1.08148e9 −3.81601
\(261\) 7.51522e7 0.261637
\(262\) −1.92855e8 −0.662485
\(263\) −3.99920e8 −1.35559 −0.677795 0.735251i \(-0.737064\pi\)
−0.677795 + 0.735251i \(0.737064\pi\)
\(264\) 1.31851e8 0.441030
\(265\) −3.42324e8 −1.13000
\(266\) 1.31039e9 4.26889
\(267\) 1.13823e8 0.365967
\(268\) 3.29407e8 1.04535
\(269\) 1.62396e7 0.0508677 0.0254338 0.999677i \(-0.491903\pi\)
0.0254338 + 0.999677i \(0.491903\pi\)
\(270\) −1.63528e8 −0.505612
\(271\) −4.27140e8 −1.30370 −0.651851 0.758347i \(-0.726007\pi\)
−0.651851 + 0.758347i \(0.726007\pi\)
\(272\) −7.91470e7 −0.238475
\(273\) −5.01652e8 −1.49222
\(274\) −8.85419e8 −2.60029
\(275\) −3.28485e8 −0.952470
\(276\) −4.95604e8 −1.41890
\(277\) −4.70787e8 −1.33090 −0.665449 0.746443i \(-0.731760\pi\)
−0.665449 + 0.746443i \(0.731760\pi\)
\(278\) 1.07940e9 3.01319
\(279\) −1.18223e8 −0.325903
\(280\) −1.32995e9 −3.62061
\(281\) −3.44673e8 −0.926692 −0.463346 0.886177i \(-0.653351\pi\)
−0.463346 + 0.886177i \(0.653351\pi\)
\(282\) 5.61061e8 1.48984
\(283\) 3.05838e8 0.802120 0.401060 0.916052i \(-0.368642\pi\)
0.401060 + 0.916052i \(0.368642\pi\)
\(284\) 7.56902e8 1.96076
\(285\) 4.94687e8 1.26582
\(286\) 5.66789e8 1.43265
\(287\) −6.29965e8 −1.57300
\(288\) −1.01977e8 −0.251552
\(289\) −1.15582e8 −0.281674
\(290\) 8.56473e8 2.06215
\(291\) 4.31797e7 0.102720
\(292\) −8.75007e8 −2.05670
\(293\) −8.35017e7 −0.193936 −0.0969680 0.995287i \(-0.530914\pi\)
−0.0969680 + 0.995287i \(0.530914\pi\)
\(294\) −1.03718e9 −2.38034
\(295\) 9.11538e7 0.206727
\(296\) 7.89728e8 1.76993
\(297\) 5.43957e7 0.120481
\(298\) 1.76995e8 0.387440
\(299\) −9.04286e8 −1.95640
\(300\) −7.13736e8 −1.52621
\(301\) 1.34035e9 2.83293
\(302\) −2.89343e8 −0.604489
\(303\) 3.74465e7 0.0773325
\(304\) −1.90305e8 −0.388502
\(305\) −9.40092e7 −0.189723
\(306\) −2.34283e8 −0.467429
\(307\) −7.34193e8 −1.44819 −0.724096 0.689699i \(-0.757743\pi\)
−0.724096 + 0.689699i \(0.757743\pi\)
\(308\) 1.04226e9 2.03259
\(309\) 3.00245e7 0.0578924
\(310\) −1.34733e9 −2.56867
\(311\) 2.34224e7 0.0441540 0.0220770 0.999756i \(-0.492972\pi\)
0.0220770 + 0.999756i \(0.492972\pi\)
\(312\) 5.22728e8 0.974394
\(313\) 4.22581e8 0.778941 0.389471 0.921039i \(-0.372658\pi\)
0.389471 + 0.921039i \(0.372658\pi\)
\(314\) 6.78799e8 1.23734
\(315\) −5.48678e8 −0.989079
\(316\) 1.01277e9 1.80553
\(317\) −1.79803e7 −0.0317023 −0.0158511 0.999874i \(-0.505046\pi\)
−0.0158511 + 0.999874i \(0.505046\pi\)
\(318\) 3.89819e8 0.679780
\(319\) −2.84897e8 −0.491383
\(320\) −1.42408e9 −2.42945
\(321\) 6.49171e8 1.09545
\(322\) −2.61994e9 −4.37317
\(323\) 7.08729e8 1.17023
\(324\) 1.18192e8 0.193054
\(325\) −1.30229e9 −2.10435
\(326\) 9.87695e8 1.57892
\(327\) −2.86958e8 −0.453838
\(328\) 6.56432e8 1.02714
\(329\) 1.88251e9 2.91442
\(330\) 6.19922e8 0.949594
\(331\) −2.19639e8 −0.332899 −0.166449 0.986050i \(-0.553230\pi\)
−0.166449 + 0.986050i \(0.553230\pi\)
\(332\) −1.00514e9 −1.50745
\(333\) 3.25807e8 0.483510
\(334\) −1.37893e8 −0.202502
\(335\) 6.57386e8 0.955352
\(336\) 2.11075e8 0.303564
\(337\) 4.32909e7 0.0616158 0.0308079 0.999525i \(-0.490192\pi\)
0.0308079 + 0.999525i \(0.490192\pi\)
\(338\) 1.07247e9 1.51070
\(339\) −4.92384e8 −0.686443
\(340\) −1.69466e9 −2.33833
\(341\) 4.48176e8 0.612080
\(342\) −5.63322e8 −0.761492
\(343\) −2.08346e9 −2.78777
\(344\) −1.39667e9 −1.84986
\(345\) −9.89058e8 −1.29675
\(346\) 1.47419e9 1.91332
\(347\) 2.07119e8 0.266114 0.133057 0.991108i \(-0.457521\pi\)
0.133057 + 0.991108i \(0.457521\pi\)
\(348\) −6.19027e8 −0.787376
\(349\) −1.35268e9 −1.70336 −0.851680 0.524062i \(-0.824416\pi\)
−0.851680 + 0.524062i \(0.824416\pi\)
\(350\) −3.77307e9 −4.70389
\(351\) 2.15654e8 0.266185
\(352\) 3.86587e8 0.472442
\(353\) −9.24750e8 −1.11896 −0.559478 0.828845i \(-0.688998\pi\)
−0.559478 + 0.828845i \(0.688998\pi\)
\(354\) −1.03801e8 −0.124362
\(355\) 1.51052e9 1.79196
\(356\) −9.37560e8 −1.10135
\(357\) −7.86081e8 −0.914384
\(358\) 3.99486e8 0.460162
\(359\) 5.95223e8 0.678968 0.339484 0.940612i \(-0.389748\pi\)
0.339484 + 0.940612i \(0.389748\pi\)
\(360\) 5.71730e8 0.645851
\(361\) 8.10232e8 0.906430
\(362\) 3.22421e8 0.357226
\(363\) 3.19943e8 0.351075
\(364\) 4.13209e9 4.49071
\(365\) −1.74622e9 −1.87964
\(366\) 1.07052e8 0.114133
\(367\) 5.60873e8 0.592289 0.296144 0.955143i \(-0.404299\pi\)
0.296144 + 0.955143i \(0.404299\pi\)
\(368\) 3.80488e8 0.397992
\(369\) 2.70815e8 0.280595
\(370\) 3.71306e9 3.81089
\(371\) 1.30795e9 1.32979
\(372\) 9.73802e8 0.980777
\(373\) −6.85459e8 −0.683912 −0.341956 0.939716i \(-0.611089\pi\)
−0.341956 + 0.939716i \(0.611089\pi\)
\(374\) 8.88150e8 0.877881
\(375\) −4.88168e8 −0.478036
\(376\) −1.96160e9 −1.90306
\(377\) −1.12949e9 −1.08564
\(378\) 6.24804e8 0.595008
\(379\) −1.03369e9 −0.975337 −0.487668 0.873029i \(-0.662152\pi\)
−0.487668 + 0.873029i \(0.662152\pi\)
\(380\) −4.07472e9 −3.80939
\(381\) −3.47236e8 −0.321653
\(382\) 4.98632e8 0.457676
\(383\) 2.09792e9 1.90806 0.954032 0.299705i \(-0.0968882\pi\)
0.954032 + 0.299705i \(0.0968882\pi\)
\(384\) 1.13821e9 1.02580
\(385\) 2.08000e9 1.85760
\(386\) −3.17582e9 −2.81061
\(387\) −5.76202e8 −0.505343
\(388\) −3.55670e8 −0.309126
\(389\) −6.60600e7 −0.0569003 −0.0284502 0.999595i \(-0.509057\pi\)
−0.0284502 + 0.999595i \(0.509057\pi\)
\(390\) 2.45771e9 2.09799
\(391\) −1.41700e9 −1.19882
\(392\) 3.62623e9 3.04056
\(393\) 2.78172e8 0.231174
\(394\) 1.20069e9 0.988998
\(395\) 2.02115e9 1.65009
\(396\) −4.48056e8 −0.362576
\(397\) 1.83981e9 1.47573 0.737863 0.674951i \(-0.235835\pi\)
0.737863 + 0.674951i \(0.235835\pi\)
\(398\) −2.14868e9 −1.70837
\(399\) −1.89009e9 −1.48963
\(400\) 5.47955e8 0.428090
\(401\) 1.61549e9 1.25112 0.625560 0.780176i \(-0.284870\pi\)
0.625560 + 0.780176i \(0.284870\pi\)
\(402\) −7.48594e8 −0.574718
\(403\) 1.77681e9 1.35230
\(404\) −3.08446e8 −0.232726
\(405\) 2.35871e8 0.176434
\(406\) −3.27240e9 −2.42675
\(407\) −1.23511e9 −0.908083
\(408\) 8.19107e8 0.597077
\(409\) −1.07871e9 −0.779605 −0.389802 0.920899i \(-0.627457\pi\)
−0.389802 + 0.920899i \(0.627457\pi\)
\(410\) 3.08634e9 2.21157
\(411\) 1.27712e9 0.907372
\(412\) −2.47311e8 −0.174222
\(413\) −3.48279e8 −0.243278
\(414\) 1.12628e9 0.780093
\(415\) −2.00591e9 −1.37767
\(416\) 1.53264e9 1.04379
\(417\) −1.55692e9 −1.05145
\(418\) 2.13551e9 1.43016
\(419\) −8.10783e8 −0.538463 −0.269231 0.963076i \(-0.586770\pi\)
−0.269231 + 0.963076i \(0.586770\pi\)
\(420\) 4.51945e9 2.97655
\(421\) −1.13878e9 −0.743792 −0.371896 0.928274i \(-0.621292\pi\)
−0.371896 + 0.928274i \(0.621292\pi\)
\(422\) 1.27192e8 0.0823886
\(423\) −8.09269e8 −0.519878
\(424\) −1.36290e9 −0.868327
\(425\) −2.04068e9 −1.28948
\(426\) −1.72010e9 −1.07800
\(427\) 3.59189e8 0.223268
\(428\) −5.34720e9 −3.29665
\(429\) −8.17530e8 −0.499923
\(430\) −6.56670e9 −3.98297
\(431\) −2.09869e9 −1.26263 −0.631317 0.775525i \(-0.717485\pi\)
−0.631317 + 0.775525i \(0.717485\pi\)
\(432\) −9.07389e7 −0.0541503
\(433\) 1.85715e9 1.09936 0.549679 0.835376i \(-0.314750\pi\)
0.549679 + 0.835376i \(0.314750\pi\)
\(434\) 5.14787e9 3.02283
\(435\) −1.23537e9 −0.719588
\(436\) 2.36366e9 1.36579
\(437\) −3.40712e9 −1.95300
\(438\) 1.98850e9 1.13075
\(439\) 6.58145e8 0.371275 0.185638 0.982618i \(-0.440565\pi\)
0.185638 + 0.982618i \(0.440565\pi\)
\(440\) −2.16739e9 −1.21298
\(441\) 1.49602e9 0.830620
\(442\) 3.52111e9 1.93955
\(443\) −2.56332e8 −0.140084 −0.0700422 0.997544i \(-0.522313\pi\)
−0.0700422 + 0.997544i \(0.522313\pi\)
\(444\) −2.68366e9 −1.45508
\(445\) −1.87105e9 −1.00653
\(446\) 5.22056e9 2.78641
\(447\) −2.55296e8 −0.135197
\(448\) 5.44110e9 2.85900
\(449\) 5.70165e8 0.297261 0.148631 0.988893i \(-0.452513\pi\)
0.148631 + 0.988893i \(0.452513\pi\)
\(450\) 1.62200e9 0.839087
\(451\) −1.02664e9 −0.526987
\(452\) 4.05575e9 2.06579
\(453\) 4.17346e8 0.210937
\(454\) 7.86086e8 0.394253
\(455\) 8.24626e9 4.10409
\(456\) 1.96950e9 0.972702
\(457\) −1.42553e8 −0.0698669 −0.0349334 0.999390i \(-0.511122\pi\)
−0.0349334 + 0.999390i \(0.511122\pi\)
\(458\) −2.82208e8 −0.137259
\(459\) 3.37927e8 0.163109
\(460\) 8.14684e9 3.90245
\(461\) 2.44186e9 1.16083 0.580414 0.814321i \(-0.302891\pi\)
0.580414 + 0.814321i \(0.302891\pi\)
\(462\) −2.36859e9 −1.11749
\(463\) −2.57981e9 −1.20796 −0.603982 0.796998i \(-0.706420\pi\)
−0.603982 + 0.796998i \(0.706420\pi\)
\(464\) 4.75244e8 0.220853
\(465\) 1.94338e9 0.896339
\(466\) 9.82495e8 0.449759
\(467\) −1.03284e9 −0.469272 −0.234636 0.972083i \(-0.575390\pi\)
−0.234636 + 0.972083i \(0.575390\pi\)
\(468\) −1.77634e9 −0.801060
\(469\) −2.51173e9 −1.12426
\(470\) −9.22285e9 −4.09753
\(471\) −9.79093e8 −0.431768
\(472\) 3.62912e8 0.158856
\(473\) 2.18434e9 0.949089
\(474\) −2.30157e9 −0.992657
\(475\) −4.90671e9 −2.10069
\(476\) 6.47493e9 2.75176
\(477\) −5.62272e8 −0.237209
\(478\) 3.41181e9 1.42885
\(479\) −2.54815e9 −1.05938 −0.529688 0.848193i \(-0.677691\pi\)
−0.529688 + 0.848193i \(0.677691\pi\)
\(480\) 1.67632e9 0.691850
\(481\) −4.89665e9 −2.00628
\(482\) −1.03187e9 −0.419719
\(483\) 3.77898e9 1.52602
\(484\) −2.63536e9 −1.05653
\(485\) −7.09797e8 −0.282513
\(486\) −2.68596e8 −0.106138
\(487\) −1.38050e9 −0.541608 −0.270804 0.962635i \(-0.587289\pi\)
−0.270804 + 0.962635i \(0.587289\pi\)
\(488\) −3.74280e8 −0.145790
\(489\) −1.42464e9 −0.550966
\(490\) 1.70494e10 6.54671
\(491\) 3.40284e9 1.29735 0.648674 0.761067i \(-0.275324\pi\)
0.648674 + 0.761067i \(0.275324\pi\)
\(492\) −2.23069e9 −0.844427
\(493\) −1.76989e9 −0.665246
\(494\) 8.46634e9 3.15974
\(495\) −8.94169e8 −0.331361
\(496\) −7.47614e8 −0.275101
\(497\) −5.77138e9 −2.10879
\(498\) 2.28422e9 0.828773
\(499\) −2.71570e9 −0.978430 −0.489215 0.872163i \(-0.662717\pi\)
−0.489215 + 0.872163i \(0.662717\pi\)
\(500\) 4.02103e9 1.43861
\(501\) 1.98895e8 0.0706630
\(502\) −6.56677e9 −2.31680
\(503\) −1.12999e9 −0.395899 −0.197950 0.980212i \(-0.563428\pi\)
−0.197950 + 0.980212i \(0.563428\pi\)
\(504\) −2.18446e9 −0.760041
\(505\) −6.15554e8 −0.212690
\(506\) −4.26966e9 −1.46510
\(507\) −1.54693e9 −0.527159
\(508\) 2.86017e9 0.967986
\(509\) 2.58498e9 0.868849 0.434425 0.900708i \(-0.356952\pi\)
0.434425 + 0.900708i \(0.356952\pi\)
\(510\) 3.85119e9 1.28558
\(511\) 6.67193e9 2.21197
\(512\) −1.68757e9 −0.555670
\(513\) 8.12530e8 0.265723
\(514\) −6.25885e9 −2.03294
\(515\) −4.93550e8 −0.159223
\(516\) 4.74616e9 1.52079
\(517\) 3.06788e9 0.976387
\(518\) −1.41868e10 −4.48467
\(519\) −2.12636e9 −0.667655
\(520\) −8.59272e9 −2.67990
\(521\) −1.68355e9 −0.521546 −0.260773 0.965400i \(-0.583977\pi\)
−0.260773 + 0.965400i \(0.583977\pi\)
\(522\) 1.40677e9 0.432888
\(523\) −2.47225e9 −0.755676 −0.377838 0.925872i \(-0.623332\pi\)
−0.377838 + 0.925872i \(0.623332\pi\)
\(524\) −2.29129e9 −0.695699
\(525\) 5.44224e9 1.64142
\(526\) −7.48608e9 −2.24287
\(527\) 2.78424e9 0.828648
\(528\) 3.43985e8 0.101700
\(529\) 3.40723e9 1.00071
\(530\) −6.40794e9 −1.86962
\(531\) 1.49721e8 0.0433963
\(532\) 1.55687e10 4.48291
\(533\) −4.07016e9 −1.16430
\(534\) 2.13065e9 0.605505
\(535\) −1.06712e10 −3.01284
\(536\) 2.61726e9 0.734125
\(537\) −5.76214e8 −0.160573
\(538\) 3.03988e8 0.0841624
\(539\) −5.67131e9 −1.55999
\(540\) −1.94286e9 −0.530962
\(541\) −3.60201e9 −0.978034 −0.489017 0.872274i \(-0.662644\pi\)
−0.489017 + 0.872274i \(0.662644\pi\)
\(542\) −7.99561e9 −2.15702
\(543\) −4.65056e8 −0.124654
\(544\) 2.40163e9 0.639602
\(545\) 4.71707e9 1.24820
\(546\) −9.39038e9 −2.46893
\(547\) 1.24077e9 0.324141 0.162071 0.986779i \(-0.448183\pi\)
0.162071 + 0.986779i \(0.448183\pi\)
\(548\) −1.05196e10 −2.73066
\(549\) −1.54411e8 −0.0398269
\(550\) −6.14889e9 −1.57590
\(551\) −4.25561e9 −1.08376
\(552\) −3.93775e9 −0.996463
\(553\) −7.72237e9 −1.94184
\(554\) −8.81262e9 −2.20202
\(555\) −5.35568e9 −1.32981
\(556\) 1.28243e10 3.16426
\(557\) 1.77982e9 0.436399 0.218200 0.975904i \(-0.429982\pi\)
0.218200 + 0.975904i \(0.429982\pi\)
\(558\) −2.21301e9 −0.539218
\(559\) 8.65993e9 2.09688
\(560\) −3.46970e9 −0.834900
\(561\) −1.28106e9 −0.306337
\(562\) −6.45192e9 −1.53325
\(563\) −6.60782e9 −1.56056 −0.780278 0.625433i \(-0.784922\pi\)
−0.780278 + 0.625433i \(0.784922\pi\)
\(564\) 6.66593e9 1.56453
\(565\) 8.09391e9 1.88794
\(566\) 5.72496e9 1.32714
\(567\) −9.01211e8 −0.207628
\(568\) 6.01386e9 1.37700
\(569\) 4.59260e9 1.04512 0.522560 0.852603i \(-0.324977\pi\)
0.522560 + 0.852603i \(0.324977\pi\)
\(570\) 9.26001e9 2.09435
\(571\) 3.76881e9 0.847183 0.423592 0.905853i \(-0.360769\pi\)
0.423592 + 0.905853i \(0.360769\pi\)
\(572\) 6.73398e9 1.50448
\(573\) −7.19221e8 −0.159706
\(574\) −1.17923e10 −2.60259
\(575\) 9.81029e9 2.15201
\(576\) −2.33907e9 −0.509993
\(577\) 3.58209e9 0.776284 0.388142 0.921600i \(-0.373117\pi\)
0.388142 + 0.921600i \(0.373117\pi\)
\(578\) −2.16357e9 −0.466040
\(579\) 4.58077e9 0.980762
\(580\) 1.01757e10 2.16554
\(581\) 7.66417e9 1.62125
\(582\) 8.08277e8 0.169953
\(583\) 2.13153e9 0.445504
\(584\) −6.95225e9 −1.44438
\(585\) −3.54497e9 −0.732095
\(586\) −1.56306e9 −0.320874
\(587\) 3.85835e8 0.0787350 0.0393675 0.999225i \(-0.487466\pi\)
0.0393675 + 0.999225i \(0.487466\pi\)
\(588\) −1.23227e10 −2.49968
\(589\) 6.69458e9 1.34996
\(590\) 1.70630e9 0.342037
\(591\) −1.73187e9 −0.345111
\(592\) 2.06032e9 0.408140
\(593\) −1.84467e9 −0.363267 −0.181634 0.983366i \(-0.558139\pi\)
−0.181634 + 0.983366i \(0.558139\pi\)
\(594\) 1.01823e9 0.199339
\(595\) 1.29218e10 2.51485
\(596\) 2.10287e9 0.406864
\(597\) 3.09924e9 0.596136
\(598\) −1.69273e10 −3.23693
\(599\) 4.66546e9 0.886953 0.443476 0.896286i \(-0.353745\pi\)
0.443476 + 0.896286i \(0.353745\pi\)
\(600\) −5.67089e9 −1.07182
\(601\) 2.24604e9 0.422043 0.211021 0.977481i \(-0.432321\pi\)
0.211021 + 0.977481i \(0.432321\pi\)
\(602\) 2.50900e10 4.68719
\(603\) 1.07976e9 0.200548
\(604\) −3.43766e9 −0.634796
\(605\) −5.25930e9 −0.965570
\(606\) 7.00958e8 0.127949
\(607\) 9.43363e9 1.71206 0.856029 0.516928i \(-0.172925\pi\)
0.856029 + 0.516928i \(0.172925\pi\)
\(608\) 5.77460e9 1.04198
\(609\) 4.72008e9 0.846816
\(610\) −1.75975e9 −0.313904
\(611\) 1.21628e10 2.15719
\(612\) −2.78350e9 −0.490864
\(613\) 2.56081e9 0.449020 0.224510 0.974472i \(-0.427922\pi\)
0.224510 + 0.974472i \(0.427922\pi\)
\(614\) −1.37433e10 −2.39608
\(615\) −4.45171e9 −0.771728
\(616\) 8.28114e9 1.42744
\(617\) 2.42876e9 0.416281 0.208141 0.978099i \(-0.433259\pi\)
0.208141 + 0.978099i \(0.433259\pi\)
\(618\) 5.62027e8 0.0957850
\(619\) 6.53273e9 1.10708 0.553538 0.832824i \(-0.313277\pi\)
0.553538 + 0.832824i \(0.313277\pi\)
\(620\) −1.60076e10 −2.69746
\(621\) −1.62454e9 −0.272214
\(622\) 4.38443e8 0.0730544
\(623\) 7.14890e9 1.18449
\(624\) 1.36374e9 0.224692
\(625\) −1.26146e9 −0.206678
\(626\) 7.91026e9 1.28879
\(627\) −3.08024e9 −0.499055
\(628\) 8.06476e9 1.29937
\(629\) −7.67299e9 −1.22938
\(630\) −1.02707e10 −1.63647
\(631\) −2.21286e9 −0.350631 −0.175315 0.984512i \(-0.556095\pi\)
−0.175315 + 0.984512i \(0.556095\pi\)
\(632\) 8.04681e9 1.26799
\(633\) −1.83461e8 −0.0287495
\(634\) −3.36573e8 −0.0524525
\(635\) 5.70794e9 0.884649
\(636\) 4.63142e9 0.713861
\(637\) −2.24842e10 −3.44658
\(638\) −5.33296e9 −0.813011
\(639\) 2.48105e9 0.376169
\(640\) −1.87102e10 −2.82130
\(641\) 5.20588e9 0.780713 0.390356 0.920664i \(-0.372352\pi\)
0.390356 + 0.920664i \(0.372352\pi\)
\(642\) 1.21518e10 1.81246
\(643\) −1.18888e10 −1.76359 −0.881796 0.471631i \(-0.843665\pi\)
−0.881796 + 0.471631i \(0.843665\pi\)
\(644\) −3.11274e10 −4.59242
\(645\) 9.47174e9 1.38986
\(646\) 1.32666e10 1.93619
\(647\) −6.02914e9 −0.875166 −0.437583 0.899178i \(-0.644165\pi\)
−0.437583 + 0.899178i \(0.644165\pi\)
\(648\) 9.39075e8 0.135577
\(649\) −5.67583e8 −0.0815029
\(650\) −2.43776e10 −3.48172
\(651\) −7.42524e9 −1.05482
\(652\) 1.17347e10 1.65809
\(653\) 2.77068e8 0.0389395 0.0194697 0.999810i \(-0.493802\pi\)
0.0194697 + 0.999810i \(0.493802\pi\)
\(654\) −5.37154e9 −0.750890
\(655\) −4.57265e9 −0.635804
\(656\) 1.71256e9 0.236856
\(657\) −2.86819e9 −0.394574
\(658\) 3.52386e10 4.82200
\(659\) −1.19887e10 −1.63182 −0.815909 0.578181i \(-0.803763\pi\)
−0.815909 + 0.578181i \(0.803763\pi\)
\(660\) 7.36525e9 0.997203
\(661\) 3.39523e9 0.457261 0.228631 0.973513i \(-0.426575\pi\)
0.228631 + 0.973513i \(0.426575\pi\)
\(662\) −4.11142e9 −0.550793
\(663\) −5.07882e9 −0.676808
\(664\) −7.98617e9 −1.05865
\(665\) 3.10698e10 4.09697
\(666\) 6.09875e9 0.799984
\(667\) 8.50851e9 1.11023
\(668\) −1.63829e9 −0.212654
\(669\) −7.53008e9 −0.972318
\(670\) 1.23056e10 1.58066
\(671\) 5.85363e8 0.0747991
\(672\) −6.40485e9 −0.814173
\(673\) 9.44306e9 1.19415 0.597077 0.802184i \(-0.296329\pi\)
0.597077 + 0.802184i \(0.296329\pi\)
\(674\) 8.10359e8 0.101945
\(675\) −2.33956e9 −0.292800
\(676\) 1.27420e10 1.58644
\(677\) 2.31127e9 0.286279 0.143140 0.989703i \(-0.454280\pi\)
0.143140 + 0.989703i \(0.454280\pi\)
\(678\) −9.21690e9 −1.13574
\(679\) 2.71199e9 0.332463
\(680\) −1.34647e10 −1.64216
\(681\) −1.13384e9 −0.137575
\(682\) 8.38938e9 1.01271
\(683\) −1.31138e10 −1.57491 −0.787454 0.616374i \(-0.788601\pi\)
−0.787454 + 0.616374i \(0.788601\pi\)
\(684\) −6.69279e9 −0.799670
\(685\) −2.09936e10 −2.49557
\(686\) −3.90002e10 −4.61246
\(687\) 4.07054e8 0.0478964
\(688\) −3.64376e9 −0.426570
\(689\) 8.45056e9 0.984279
\(690\) −1.85141e10 −2.14551
\(691\) 8.33408e9 0.960913 0.480457 0.877018i \(-0.340471\pi\)
0.480457 + 0.877018i \(0.340471\pi\)
\(692\) 1.75148e10 2.00925
\(693\) 3.41643e9 0.389948
\(694\) 3.87705e9 0.440295
\(695\) 2.55930e10 2.89184
\(696\) −4.91839e9 −0.552956
\(697\) −6.37789e9 −0.713448
\(698\) −2.53207e10 −2.81827
\(699\) −1.41714e9 −0.156943
\(700\) −4.48276e10 −4.93972
\(701\) −2.09612e9 −0.229829 −0.114914 0.993375i \(-0.536659\pi\)
−0.114914 + 0.993375i \(0.536659\pi\)
\(702\) 4.03682e9 0.440412
\(703\) −1.84493e10 −2.00280
\(704\) 8.86724e9 0.957821
\(705\) 1.33029e10 1.42984
\(706\) −1.73103e10 −1.85135
\(707\) 2.35190e9 0.250294
\(708\) −1.23325e9 −0.130597
\(709\) 4.17621e8 0.0440069 0.0220035 0.999758i \(-0.492996\pi\)
0.0220035 + 0.999758i \(0.492996\pi\)
\(710\) 2.82753e10 2.96486
\(711\) 3.31976e9 0.346388
\(712\) −7.44925e9 −0.773451
\(713\) −1.33849e10 −1.38293
\(714\) −1.47146e10 −1.51288
\(715\) 1.34387e10 1.37495
\(716\) 4.74626e9 0.483232
\(717\) −4.92116e9 −0.498598
\(718\) 1.11419e10 1.12338
\(719\) −6.71535e9 −0.673779 −0.336890 0.941544i \(-0.609375\pi\)
−0.336890 + 0.941544i \(0.609375\pi\)
\(720\) 1.49159e9 0.148931
\(721\) 1.88575e9 0.187375
\(722\) 1.51667e10 1.49972
\(723\) 1.48835e9 0.146461
\(724\) 3.83066e9 0.375136
\(725\) 1.22534e10 1.19419
\(726\) 5.98899e9 0.580866
\(727\) −8.79100e9 −0.848531 −0.424266 0.905538i \(-0.639468\pi\)
−0.424266 + 0.905538i \(0.639468\pi\)
\(728\) 3.28309e10 3.15372
\(729\) 3.87420e8 0.0370370
\(730\) −3.26874e10 −3.10992
\(731\) 1.35700e10 1.28490
\(732\) 1.27188e9 0.119856
\(733\) −1.87568e10 −1.75912 −0.879560 0.475787i \(-0.842163\pi\)
−0.879560 + 0.475787i \(0.842163\pi\)
\(734\) 1.04989e10 0.979963
\(735\) −2.45919e10 −2.28448
\(736\) −1.15455e10 −1.06743
\(737\) −4.09331e9 −0.376651
\(738\) 5.06936e9 0.464254
\(739\) −3.83197e9 −0.349274 −0.174637 0.984633i \(-0.555875\pi\)
−0.174637 + 0.984633i \(0.555875\pi\)
\(740\) 4.41146e10 4.00195
\(741\) −1.22118e10 −1.10259
\(742\) 2.44834e10 2.20018
\(743\) −1.74075e9 −0.155695 −0.0778475 0.996965i \(-0.524805\pi\)
−0.0778475 + 0.996965i \(0.524805\pi\)
\(744\) 7.73721e9 0.688777
\(745\) 4.19661e9 0.371836
\(746\) −1.28311e10 −1.13156
\(747\) −3.29474e9 −0.289201
\(748\) 1.05521e10 0.921895
\(749\) 4.07724e10 3.54552
\(750\) −9.13799e9 −0.790927
\(751\) 1.87554e10 1.61580 0.807900 0.589320i \(-0.200604\pi\)
0.807900 + 0.589320i \(0.200604\pi\)
\(752\) −5.11762e9 −0.438839
\(753\) 9.47185e9 0.808448
\(754\) −2.11428e10 −1.79623
\(755\) −6.86042e9 −0.580145
\(756\) 7.42325e9 0.624839
\(757\) −1.48636e10 −1.24534 −0.622671 0.782483i \(-0.713953\pi\)
−0.622671 + 0.782483i \(0.713953\pi\)
\(758\) −1.93496e10 −1.61373
\(759\) 6.15852e9 0.511246
\(760\) −3.23751e10 −2.67525
\(761\) 2.55852e9 0.210446 0.105223 0.994449i \(-0.466444\pi\)
0.105223 + 0.994449i \(0.466444\pi\)
\(762\) −6.49988e9 −0.532185
\(763\) −1.80229e10 −1.46889
\(764\) 5.92421e9 0.480622
\(765\) −5.55492e9 −0.448604
\(766\) 3.92708e10 3.15696
\(767\) −2.25021e9 −0.180069
\(768\) 1.02172e10 0.813897
\(769\) 1.98268e10 1.57221 0.786104 0.618094i \(-0.212095\pi\)
0.786104 + 0.618094i \(0.212095\pi\)
\(770\) 3.89354e10 3.07346
\(771\) 9.02771e9 0.709393
\(772\) −3.77317e10 −2.95152
\(773\) −1.29417e10 −1.00777 −0.503886 0.863770i \(-0.668097\pi\)
−0.503886 + 0.863770i \(0.668097\pi\)
\(774\) −1.07859e10 −0.836109
\(775\) −1.92760e10 −1.48752
\(776\) −2.82593e9 −0.217092
\(777\) 2.04629e10 1.56493
\(778\) −1.23657e9 −0.0941436
\(779\) −1.53353e10 −1.16228
\(780\) 2.91998e10 2.20318
\(781\) −9.40549e9 −0.706485
\(782\) −2.65248e10 −1.98348
\(783\) −2.02911e9 −0.151056
\(784\) 9.46046e9 0.701142
\(785\) 1.60945e10 1.18750
\(786\) 5.20708e9 0.382486
\(787\) 9.60328e9 0.702276 0.351138 0.936324i \(-0.385795\pi\)
0.351138 + 0.936324i \(0.385795\pi\)
\(788\) 1.42654e10 1.03858
\(789\) 1.07978e10 0.782650
\(790\) 3.78337e10 2.73013
\(791\) −3.09251e10 −2.22174
\(792\) −3.55997e9 −0.254629
\(793\) 2.32070e9 0.165258
\(794\) 3.44392e10 2.44164
\(795\) 9.24274e9 0.652403
\(796\) −2.55284e10 −1.79402
\(797\) −9.10764e9 −0.637238 −0.318619 0.947883i \(-0.603219\pi\)
−0.318619 + 0.947883i \(0.603219\pi\)
\(798\) −3.53805e10 −2.46464
\(799\) 1.90589e10 1.32185
\(800\) −1.66271e10 −1.14816
\(801\) −3.07323e9 −0.211291
\(802\) 3.02403e10 2.07002
\(803\) 1.08731e10 0.741053
\(804\) −8.89399e9 −0.603532
\(805\) −6.21197e10 −4.19705
\(806\) 3.32601e10 2.23744
\(807\) −4.38469e8 −0.0293685
\(808\) −2.45071e9 −0.163438
\(809\) −1.78494e10 −1.18524 −0.592618 0.805484i \(-0.701906\pi\)
−0.592618 + 0.805484i \(0.701906\pi\)
\(810\) 4.41524e9 0.291915
\(811\) 6.82320e9 0.449174 0.224587 0.974454i \(-0.427897\pi\)
0.224587 + 0.974454i \(0.427897\pi\)
\(812\) −3.88792e10 −2.54842
\(813\) 1.15328e10 0.752692
\(814\) −2.31200e10 −1.50246
\(815\) 2.34186e10 1.51534
\(816\) 2.13697e9 0.137684
\(817\) 3.26284e10 2.09324
\(818\) −2.01924e10 −1.28988
\(819\) 1.35446e10 0.861534
\(820\) 3.66686e10 2.32245
\(821\) 7.07763e9 0.446362 0.223181 0.974777i \(-0.428356\pi\)
0.223181 + 0.974777i \(0.428356\pi\)
\(822\) 2.39063e10 1.50128
\(823\) −1.02079e10 −0.638317 −0.319159 0.947701i \(-0.603400\pi\)
−0.319159 + 0.947701i \(0.603400\pi\)
\(824\) −1.96498e9 −0.122352
\(825\) 8.86910e9 0.549909
\(826\) −6.51941e9 −0.402512
\(827\) 2.59808e9 0.159729 0.0798644 0.996806i \(-0.474551\pi\)
0.0798644 + 0.996806i \(0.474551\pi\)
\(828\) 1.33813e10 0.819204
\(829\) 2.93686e9 0.179037 0.0895184 0.995985i \(-0.471467\pi\)
0.0895184 + 0.995985i \(0.471467\pi\)
\(830\) −3.75486e10 −2.27940
\(831\) 1.27112e10 0.768395
\(832\) 3.51546e10 2.11617
\(833\) −3.52324e10 −2.11195
\(834\) −2.91439e10 −1.73967
\(835\) −3.26948e9 −0.194346
\(836\) 2.53719e10 1.50187
\(837\) 3.19203e9 0.188160
\(838\) −1.51770e10 −0.890905
\(839\) 1.16450e10 0.680728 0.340364 0.940294i \(-0.389450\pi\)
0.340364 + 0.940294i \(0.389450\pi\)
\(840\) 3.59086e10 2.09036
\(841\) −6.62244e9 −0.383912
\(842\) −2.13167e10 −1.23063
\(843\) 9.30618e9 0.535026
\(844\) 1.51116e9 0.0865192
\(845\) 2.54287e10 1.44986
\(846\) −1.51487e10 −0.860157
\(847\) 2.00947e10 1.13629
\(848\) −3.55567e9 −0.200233
\(849\) −8.25763e9 −0.463104
\(850\) −3.81993e10 −2.13348
\(851\) 3.68869e10 2.05172
\(852\) −2.04363e10 −1.13205
\(853\) 1.63671e10 0.902922 0.451461 0.892291i \(-0.350903\pi\)
0.451461 + 0.892291i \(0.350903\pi\)
\(854\) 6.72364e9 0.369404
\(855\) −1.33565e10 −0.730824
\(856\) −4.24854e10 −2.31517
\(857\) −1.66058e10 −0.901210 −0.450605 0.892724i \(-0.648792\pi\)
−0.450605 + 0.892724i \(0.648792\pi\)
\(858\) −1.53033e10 −0.827141
\(859\) −1.88831e10 −1.01647 −0.508237 0.861217i \(-0.669703\pi\)
−0.508237 + 0.861217i \(0.669703\pi\)
\(860\) −7.80185e10 −4.18266
\(861\) 1.70091e10 0.908174
\(862\) −3.92852e10 −2.08907
\(863\) 2.09759e10 1.11092 0.555461 0.831543i \(-0.312542\pi\)
0.555461 + 0.831543i \(0.312542\pi\)
\(864\) 2.75337e9 0.145234
\(865\) 3.49536e10 1.83627
\(866\) 3.47639e10 1.81893
\(867\) 3.12071e9 0.162625
\(868\) 6.11615e10 3.17438
\(869\) −1.25850e10 −0.650554
\(870\) −2.31248e10 −1.19058
\(871\) −1.62281e10 −0.832156
\(872\) 1.87801e10 0.959161
\(873\) −1.16585e9 −0.0593053
\(874\) −6.37776e10 −3.23131
\(875\) −3.06604e10 −1.54721
\(876\) 2.36252e10 1.18744
\(877\) 3.68896e10 1.84674 0.923370 0.383911i \(-0.125423\pi\)
0.923370 + 0.383911i \(0.125423\pi\)
\(878\) 1.23198e10 0.614288
\(879\) 2.25455e9 0.111969
\(880\) −5.65450e9 −0.279708
\(881\) −3.57144e10 −1.75966 −0.879829 0.475291i \(-0.842343\pi\)
−0.879829 + 0.475291i \(0.842343\pi\)
\(882\) 2.80039e10 1.37429
\(883\) −3.90952e9 −0.191100 −0.0955501 0.995425i \(-0.530461\pi\)
−0.0955501 + 0.995425i \(0.530461\pi\)
\(884\) 4.18341e10 2.03679
\(885\) −2.46115e9 −0.119354
\(886\) −4.79826e9 −0.231775
\(887\) −1.25223e10 −0.602494 −0.301247 0.953546i \(-0.597403\pi\)
−0.301247 + 0.953546i \(0.597403\pi\)
\(888\) −2.13227e10 −1.02187
\(889\) −2.18088e10 −1.04106
\(890\) −3.50241e10 −1.66534
\(891\) −1.46868e9 −0.0695595
\(892\) 6.20251e10 2.92611
\(893\) 4.58262e10 2.15344
\(894\) −4.77887e9 −0.223688
\(895\) 9.47193e9 0.441630
\(896\) 7.14878e10 3.32012
\(897\) 2.44157e10 1.12953
\(898\) 1.06729e10 0.491829
\(899\) −1.67182e10 −0.767416
\(900\) 1.92709e10 0.881156
\(901\) 1.32419e10 0.603134
\(902\) −1.92176e10 −0.871919
\(903\) −3.61895e10 −1.63559
\(904\) 3.22244e10 1.45076
\(905\) 7.64470e9 0.342839
\(906\) 7.81226e9 0.349002
\(907\) 2.27291e10 1.01148 0.505739 0.862687i \(-0.331220\pi\)
0.505739 + 0.862687i \(0.331220\pi\)
\(908\) 9.33944e9 0.414019
\(909\) −1.01106e9 −0.0446479
\(910\) 1.54361e11 6.79036
\(911\) −2.77093e10 −1.21426 −0.607129 0.794603i \(-0.707679\pi\)
−0.607129 + 0.794603i \(0.707679\pi\)
\(912\) 5.13824e9 0.224301
\(913\) 1.24901e10 0.543150
\(914\) −2.66845e9 −0.115597
\(915\) 2.53825e9 0.109537
\(916\) −3.35289e9 −0.144140
\(917\) 1.74711e10 0.748218
\(918\) 6.32564e9 0.269870
\(919\) −1.24377e10 −0.528612 −0.264306 0.964439i \(-0.585143\pi\)
−0.264306 + 0.964439i \(0.585143\pi\)
\(920\) 6.47296e10 2.74060
\(921\) 1.98232e10 0.836114
\(922\) 4.57091e10 1.92063
\(923\) −3.72885e10 −1.56088
\(924\) −2.81410e10 −1.17351
\(925\) 5.31221e10 2.20688
\(926\) −4.82913e10 −1.99862
\(927\) −8.10663e8 −0.0334242
\(928\) −1.44208e10 −0.592339
\(929\) −3.14739e10 −1.28794 −0.643970 0.765051i \(-0.722714\pi\)
−0.643970 + 0.765051i \(0.722714\pi\)
\(930\) 3.63780e10 1.48302
\(931\) −8.47146e10 −3.44060
\(932\) 1.16730e10 0.472308
\(933\) −6.32405e8 −0.0254923
\(934\) −1.93337e10 −0.776428
\(935\) 2.10583e10 0.842526
\(936\) −1.41136e10 −0.562567
\(937\) −3.85637e10 −1.53140 −0.765702 0.643195i \(-0.777608\pi\)
−0.765702 + 0.643195i \(0.777608\pi\)
\(938\) −4.70169e10 −1.86013
\(939\) −1.14097e10 −0.449722
\(940\) −1.09576e11 −4.30297
\(941\) 2.15748e10 0.844078 0.422039 0.906578i \(-0.361314\pi\)
0.422039 + 0.906578i \(0.361314\pi\)
\(942\) −1.83276e10 −0.714376
\(943\) 3.06608e10 1.19067
\(944\) 9.46800e8 0.0366316
\(945\) 1.48143e10 0.571045
\(946\) 4.08886e10 1.57030
\(947\) −2.58130e10 −0.987675 −0.493838 0.869554i \(-0.664406\pi\)
−0.493838 + 0.869554i \(0.664406\pi\)
\(948\) −2.73448e10 −1.04243
\(949\) 4.31069e10 1.63725
\(950\) −9.18484e10 −3.47567
\(951\) 4.85469e8 0.0183033
\(952\) 5.14457e10 1.93250
\(953\) 3.92406e10 1.46862 0.734311 0.678813i \(-0.237505\pi\)
0.734311 + 0.678813i \(0.237505\pi\)
\(954\) −1.05251e10 −0.392471
\(955\) 1.18227e10 0.439244
\(956\) 4.05355e10 1.50049
\(957\) 7.69222e9 0.283700
\(958\) −4.76986e10 −1.75278
\(959\) 8.02121e10 2.93680
\(960\) 3.84501e10 1.40265
\(961\) −1.21292e9 −0.0440859
\(962\) −9.16601e10 −3.31946
\(963\) −1.75276e10 −0.632457
\(964\) −1.22595e10 −0.440762
\(965\) −7.52997e10 −2.69741
\(966\) 7.07385e10 2.52485
\(967\) −1.78901e9 −0.0636240 −0.0318120 0.999494i \(-0.510128\pi\)
−0.0318120 + 0.999494i \(0.510128\pi\)
\(968\) −2.09389e10 −0.741977
\(969\) −1.91357e10 −0.675633
\(970\) −1.32866e10 −0.467427
\(971\) −2.44481e10 −0.856993 −0.428497 0.903543i \(-0.640957\pi\)
−0.428497 + 0.903543i \(0.640957\pi\)
\(972\) −3.19117e9 −0.111460
\(973\) −9.77854e10 −3.40313
\(974\) −2.58415e10 −0.896109
\(975\) 3.51619e10 1.21495
\(976\) −9.76459e8 −0.0336186
\(977\) 3.42307e10 1.17432 0.587158 0.809472i \(-0.300247\pi\)
0.587158 + 0.809472i \(0.300247\pi\)
\(978\) −2.66678e10 −0.911593
\(979\) 1.16504e10 0.396827
\(980\) 2.02563e11 6.87494
\(981\) 7.74785e9 0.262023
\(982\) 6.36975e10 2.14651
\(983\) −4.53628e10 −1.52322 −0.761610 0.648036i \(-0.775591\pi\)
−0.761610 + 0.648036i \(0.775591\pi\)
\(984\) −1.77237e10 −0.593022
\(985\) 2.84688e10 0.949167
\(986\) −3.31304e10 −1.10067
\(987\) −5.08277e10 −1.68264
\(988\) 1.00588e11 3.31816
\(989\) −6.52359e10 −2.14437
\(990\) −1.67379e10 −0.548248
\(991\) 2.84050e10 0.927123 0.463561 0.886065i \(-0.346571\pi\)
0.463561 + 0.886065i \(0.346571\pi\)
\(992\) 2.26855e10 0.737834
\(993\) 5.93027e9 0.192199
\(994\) −1.08034e11 −3.48906
\(995\) −5.09460e10 −1.63957
\(996\) 2.71387e10 0.870324
\(997\) −3.63738e10 −1.16240 −0.581201 0.813760i \(-0.697417\pi\)
−0.581201 + 0.813760i \(0.697417\pi\)
\(998\) −5.08350e10 −1.61885
\(999\) −8.79678e9 −0.279154
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 177.8.a.a.1.15 16
3.2 odd 2 531.8.a.b.1.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.15 16 1.1 even 1 trivial
531.8.a.b.1.2 16 3.2 odd 2