Properties

Label 531.8.a.b.1.2
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
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: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-18.7189\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-18.7189 q^{2} +222.398 q^{4} -443.832 q^{5} -1695.79 q^{7} -1767.03 q^{8} +O(q^{10})\) \(q-18.7189 q^{2} +222.398 q^{4} -443.832 q^{5} -1695.79 q^{7} -1767.03 q^{8} +8308.06 q^{10} +2763.59 q^{11} -10956.4 q^{13} +31743.3 q^{14} +4610.01 q^{16} +17168.5 q^{17} -41280.8 q^{19} -98707.5 q^{20} -51731.4 q^{22} -82535.2 q^{23} +118862. q^{25} +205092. q^{26} -377140. q^{28} -103089. q^{29} -162172. q^{31} +139886. q^{32} -321376. q^{34} +752645. q^{35} +446923. q^{37} +772732. q^{38} +784266. q^{40} -371488. q^{41} -790401. q^{43} +614617. q^{44} +1.54497e6 q^{46} +1.11011e6 q^{47} +2.05215e6 q^{49} -2.22497e6 q^{50} -2.43668e6 q^{52} +771292. q^{53} -1.22657e6 q^{55} +2.99652e6 q^{56} +1.92972e6 q^{58} -205379. q^{59} -211813. q^{61} +3.03568e6 q^{62} -3.20860e6 q^{64} +4.86279e6 q^{65} +1.48116e6 q^{67} +3.81824e6 q^{68} -1.40887e7 q^{70} -3.40336e6 q^{71} -3.93442e6 q^{73} -8.36592e6 q^{74} -9.18078e6 q^{76} -4.68646e6 q^{77} +4.55385e6 q^{79} -2.04607e6 q^{80} +6.95386e6 q^{82} +4.51953e6 q^{83} -7.61992e6 q^{85} +1.47955e7 q^{86} -4.88336e6 q^{88} +4.21568e6 q^{89} +1.85797e7 q^{91} -1.83557e7 q^{92} -2.07800e7 q^{94} +1.83217e7 q^{95} -1.59925e6 q^{97} -3.84141e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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 \(-0.810106\pi\)
−0.827268 + 0.561808i \(0.810106\pi\)
\(3\) 0 0
\(4\) 222.398 1.73749
\(5\) −443.832 −1.58790 −0.793951 0.607982i \(-0.791979\pi\)
−0.793951 + 0.607982i \(0.791979\pi\)
\(6\) 0 0
\(7\) −1695.79 −1.86865 −0.934326 0.356419i \(-0.883997\pi\)
−0.934326 + 0.356419i \(0.883997\pi\)
\(8\) −1767.03 −1.22020
\(9\) 0 0
\(10\) 8308.06 2.62724
\(11\) 2763.59 0.626036 0.313018 0.949747i \(-0.398660\pi\)
0.313018 + 0.949747i \(0.398660\pi\)
\(12\) 0 0
\(13\) −10956.4 −1.38314 −0.691568 0.722311i \(-0.743080\pi\)
−0.691568 + 0.722311i \(0.743080\pi\)
\(14\) 31743.3 3.09175
\(15\) 0 0
\(16\) 4610.01 0.281373
\(17\) 17168.5 0.847541 0.423771 0.905770i \(-0.360706\pi\)
0.423771 + 0.905770i \(0.360706\pi\)
\(18\) 0 0
\(19\) −41280.8 −1.38074 −0.690368 0.723459i \(-0.742551\pi\)
−0.690368 + 0.723459i \(0.742551\pi\)
\(20\) −98707.5 −2.75896
\(21\) 0 0
\(22\) −51731.4 −1.03580
\(23\) −82535.2 −1.41446 −0.707232 0.706982i \(-0.750056\pi\)
−0.707232 + 0.706982i \(0.750056\pi\)
\(24\) 0 0
\(25\) 118862. 1.52143
\(26\) 205092. 2.28845
\(27\) 0 0
\(28\) −377140. −3.24676
\(29\) −103089. −0.784912 −0.392456 0.919771i \(-0.628375\pi\)
−0.392456 + 0.919771i \(0.628375\pi\)
\(30\) 0 0
\(31\) −162172. −0.977709 −0.488854 0.872365i \(-0.662585\pi\)
−0.488854 + 0.872365i \(0.662585\pi\)
\(32\) 139886. 0.754656
\(33\) 0 0
\(34\) −321376. −1.40229
\(35\) 752645. 2.96724
\(36\) 0 0
\(37\) 446923. 1.45053 0.725265 0.688470i \(-0.241717\pi\)
0.725265 + 0.688470i \(0.241717\pi\)
\(38\) 772732. 2.28448
\(39\) 0 0
\(40\) 784266. 1.93755
\(41\) −371488. −0.841785 −0.420893 0.907111i \(-0.638283\pi\)
−0.420893 + 0.907111i \(0.638283\pi\)
\(42\) 0 0
\(43\) −790401. −1.51603 −0.758015 0.652237i \(-0.773831\pi\)
−0.758015 + 0.652237i \(0.773831\pi\)
\(44\) 614617. 1.08773
\(45\) 0 0
\(46\) 1.54497e6 2.34028
\(47\) 1.11011e6 1.55964 0.779818 0.626007i \(-0.215312\pi\)
0.779818 + 0.626007i \(0.215312\pi\)
\(48\) 0 0
\(49\) 2.05215e6 2.49186
\(50\) −2.22497e6 −2.51726
\(51\) 0 0
\(52\) −2.43668e6 −2.40318
\(53\) 771292. 0.711628 0.355814 0.934557i \(-0.384204\pi\)
0.355814 + 0.934557i \(0.384204\pi\)
\(54\) 0 0
\(55\) −1.22657e6 −0.994083
\(56\) 2.99652e6 2.28012
\(57\) 0 0
\(58\) 1.92972e6 1.29867
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −211813. −0.119481 −0.0597403 0.998214i \(-0.519027\pi\)
−0.0597403 + 0.998214i \(0.519027\pi\)
\(62\) 3.03568e6 1.61765
\(63\) 0 0
\(64\) −3.20860e6 −1.52998
\(65\) 4.86279e6 2.19628
\(66\) 0 0
\(67\) 1.48116e6 0.601644 0.300822 0.953680i \(-0.402739\pi\)
0.300822 + 0.953680i \(0.402739\pi\)
\(68\) 3.81824e6 1.47259
\(69\) 0 0
\(70\) −1.40887e7 −4.90940
\(71\) −3.40336e6 −1.12851 −0.564253 0.825602i \(-0.690836\pi\)
−0.564253 + 0.825602i \(0.690836\pi\)
\(72\) 0 0
\(73\) −3.93442e6 −1.18372 −0.591862 0.806040i \(-0.701607\pi\)
−0.591862 + 0.806040i \(0.701607\pi\)
\(74\) −8.36592e6 −2.39995
\(75\) 0 0
\(76\) −9.18078e6 −2.39901
\(77\) −4.68646e6 −1.16984
\(78\) 0 0
\(79\) 4.55385e6 1.03916 0.519582 0.854421i \(-0.326088\pi\)
0.519582 + 0.854421i \(0.326088\pi\)
\(80\) −2.04607e6 −0.446793
\(81\) 0 0
\(82\) 6.95386e6 1.39276
\(83\) 4.51953e6 0.867602 0.433801 0.901009i \(-0.357172\pi\)
0.433801 + 0.901009i \(0.357172\pi\)
\(84\) 0 0
\(85\) −7.61992e6 −1.34581
\(86\) 1.47955e7 2.50833
\(87\) 0 0
\(88\) −4.88336e6 −0.763887
\(89\) 4.21568e6 0.633874 0.316937 0.948447i \(-0.397346\pi\)
0.316937 + 0.948447i \(0.397346\pi\)
\(90\) 0 0
\(91\) 1.85797e7 2.58460
\(92\) −1.83557e7 −2.45761
\(93\) 0 0
\(94\) −2.07800e7 −2.58047
\(95\) 1.83217e7 2.19247
\(96\) 0 0
\(97\) −1.59925e6 −0.177916 −0.0889579 0.996035i \(-0.528354\pi\)
−0.0889579 + 0.996035i \(0.528354\pi\)
\(98\) −3.84141e7 −4.12287
\(99\) 0 0
\(100\) 2.64347e7 2.64347
\(101\) 1.38691e6 0.133944 0.0669719 0.997755i \(-0.478666\pi\)
0.0669719 + 0.997755i \(0.478666\pi\)
\(102\) 0 0
\(103\) −1.11202e6 −0.100273 −0.0501363 0.998742i \(-0.515966\pi\)
−0.0501363 + 0.998742i \(0.515966\pi\)
\(104\) 1.93603e7 1.68770
\(105\) 0 0
\(106\) −1.44378e7 −1.17741
\(107\) 2.40434e7 1.89737 0.948685 0.316223i \(-0.102415\pi\)
0.948685 + 0.316223i \(0.102415\pi\)
\(108\) 0 0
\(109\) 1.06281e7 0.786070 0.393035 0.919524i \(-0.371425\pi\)
0.393035 + 0.919524i \(0.371425\pi\)
\(110\) 2.29601e7 1.64475
\(111\) 0 0
\(112\) −7.81761e6 −0.525788
\(113\) −1.82364e7 −1.18895 −0.594477 0.804112i \(-0.702641\pi\)
−0.594477 + 0.804112i \(0.702641\pi\)
\(114\) 0 0
\(115\) 3.66318e7 2.24603
\(116\) −2.29269e7 −1.36377
\(117\) 0 0
\(118\) 3.84447e6 0.215402
\(119\) −2.91141e7 −1.58376
\(120\) 0 0
\(121\) −1.18497e7 −0.608079
\(122\) 3.96491e6 0.197685
\(123\) 0 0
\(124\) −3.60667e7 −1.69876
\(125\) −1.80803e7 −0.827982
\(126\) 0 0
\(127\) 1.28606e7 0.557119 0.278559 0.960419i \(-0.410143\pi\)
0.278559 + 0.960419i \(0.410143\pi\)
\(128\) 4.21561e7 1.77675
\(129\) 0 0
\(130\) −9.10262e7 −3.63383
\(131\) 1.03027e7 0.400405 0.200203 0.979754i \(-0.435840\pi\)
0.200203 + 0.979754i \(0.435840\pi\)
\(132\) 0 0
\(133\) 7.00035e7 2.58011
\(134\) −2.77257e7 −0.995442
\(135\) 0 0
\(136\) −3.03373e7 −1.03417
\(137\) 4.73008e7 1.57161 0.785807 0.618471i \(-0.212248\pi\)
0.785807 + 0.618471i \(0.212248\pi\)
\(138\) 0 0
\(139\) 5.76637e7 1.82117 0.910585 0.413322i \(-0.135631\pi\)
0.910585 + 0.413322i \(0.135631\pi\)
\(140\) 1.67387e8 5.15553
\(141\) 0 0
\(142\) 6.37073e7 1.86715
\(143\) −3.02789e7 −0.865893
\(144\) 0 0
\(145\) 4.57544e7 1.24636
\(146\) 7.36480e7 1.95851
\(147\) 0 0
\(148\) 9.93949e7 2.52028
\(149\) −9.45541e6 −0.234168 −0.117084 0.993122i \(-0.537355\pi\)
−0.117084 + 0.993122i \(0.537355\pi\)
\(150\) 0 0
\(151\) −1.54572e7 −0.365353 −0.182677 0.983173i \(-0.558476\pi\)
−0.182677 + 0.983173i \(0.558476\pi\)
\(152\) 7.29446e7 1.68477
\(153\) 0 0
\(154\) 8.77255e7 1.93555
\(155\) 7.19770e7 1.55251
\(156\) 0 0
\(157\) 3.62627e7 0.747845 0.373922 0.927460i \(-0.378013\pi\)
0.373922 + 0.927460i \(0.378013\pi\)
\(158\) −8.52432e7 −1.71933
\(159\) 0 0
\(160\) −6.20858e7 −1.19832
\(161\) 1.39962e8 2.64314
\(162\) 0 0
\(163\) 5.27645e7 0.954301 0.477151 0.878822i \(-0.341670\pi\)
0.477151 + 0.878822i \(0.341670\pi\)
\(164\) −8.26183e7 −1.46259
\(165\) 0 0
\(166\) −8.46008e7 −1.43548
\(167\) 7.36648e6 0.122392 0.0611959 0.998126i \(-0.480509\pi\)
0.0611959 + 0.998126i \(0.480509\pi\)
\(168\) 0 0
\(169\) 5.72936e7 0.913066
\(170\) 1.42637e8 2.22669
\(171\) 0 0
\(172\) −1.75784e8 −2.63408
\(173\) −7.87542e7 −1.15641 −0.578206 0.815891i \(-0.696247\pi\)
−0.578206 + 0.815891i \(0.696247\pi\)
\(174\) 0 0
\(175\) −2.01564e8 −2.84303
\(176\) 1.27402e7 0.176149
\(177\) 0 0
\(178\) −7.89130e7 −1.04877
\(179\) −2.13413e7 −0.278121 −0.139061 0.990284i \(-0.544408\pi\)
−0.139061 + 0.990284i \(0.544408\pi\)
\(180\) 0 0
\(181\) 1.72243e7 0.215907 0.107953 0.994156i \(-0.465570\pi\)
0.107953 + 0.994156i \(0.465570\pi\)
\(182\) −3.47792e8 −4.27631
\(183\) 0 0
\(184\) 1.45843e8 1.72592
\(185\) −1.98359e8 −2.30330
\(186\) 0 0
\(187\) 4.74466e7 0.530591
\(188\) 2.46886e8 2.70984
\(189\) 0 0
\(190\) −3.42963e8 −3.62752
\(191\) −2.66378e7 −0.276619 −0.138309 0.990389i \(-0.544167\pi\)
−0.138309 + 0.990389i \(0.544167\pi\)
\(192\) 0 0
\(193\) −1.69658e8 −1.69873 −0.849365 0.527807i \(-0.823015\pi\)
−0.849365 + 0.527807i \(0.823015\pi\)
\(194\) 2.99362e7 0.294368
\(195\) 0 0
\(196\) 4.56396e8 4.32957
\(197\) −6.41433e7 −0.597750 −0.298875 0.954292i \(-0.596611\pi\)
−0.298875 + 0.954292i \(0.596611\pi\)
\(198\) 0 0
\(199\) −1.14787e8 −1.03254 −0.516269 0.856427i \(-0.672679\pi\)
−0.516269 + 0.856427i \(0.672679\pi\)
\(200\) −2.10033e8 −1.85645
\(201\) 0 0
\(202\) −2.59614e7 −0.221615
\(203\) 1.74818e8 1.46673
\(204\) 0 0
\(205\) 1.64878e8 1.33667
\(206\) 2.08158e7 0.165905
\(207\) 0 0
\(208\) −5.05090e7 −0.389177
\(209\) −1.14083e8 −0.864389
\(210\) 0 0
\(211\) 6.79484e6 0.0497956 0.0248978 0.999690i \(-0.492074\pi\)
0.0248978 + 0.999690i \(0.492074\pi\)
\(212\) 1.71534e8 1.23644
\(213\) 0 0
\(214\) −4.50066e8 −3.13926
\(215\) 3.50805e8 2.40731
\(216\) 0 0
\(217\) 2.75009e8 1.82700
\(218\) −1.98946e8 −1.30058
\(219\) 0 0
\(220\) −2.72787e8 −1.72721
\(221\) −1.88104e8 −1.17227
\(222\) 0 0
\(223\) 2.78892e8 1.68410 0.842052 0.539396i \(-0.181347\pi\)
0.842052 + 0.539396i \(0.181347\pi\)
\(224\) −2.37217e8 −1.41019
\(225\) 0 0
\(226\) 3.41367e8 1.96717
\(227\) −4.19942e7 −0.238286 −0.119143 0.992877i \(-0.538015\pi\)
−0.119143 + 0.992877i \(0.538015\pi\)
\(228\) 0 0
\(229\) −1.50761e7 −0.0829591 −0.0414795 0.999139i \(-0.513207\pi\)
−0.0414795 + 0.999139i \(0.513207\pi\)
\(230\) −6.85707e8 −3.71613
\(231\) 0 0
\(232\) 1.82163e8 0.957748
\(233\) −5.24867e7 −0.271834 −0.135917 0.990720i \(-0.543398\pi\)
−0.135917 + 0.990720i \(0.543398\pi\)
\(234\) 0 0
\(235\) −4.92702e8 −2.47655
\(236\) −4.56759e7 −0.226201
\(237\) 0 0
\(238\) 5.44985e8 2.62039
\(239\) −1.82265e8 −0.863597 −0.431799 0.901970i \(-0.642121\pi\)
−0.431799 + 0.901970i \(0.642121\pi\)
\(240\) 0 0
\(241\) −5.51242e7 −0.253678 −0.126839 0.991923i \(-0.540483\pi\)
−0.126839 + 0.991923i \(0.540483\pi\)
\(242\) 2.21815e8 1.00609
\(243\) 0 0
\(244\) −4.71068e7 −0.207596
\(245\) −9.10812e8 −3.95683
\(246\) 0 0
\(247\) 4.52288e8 1.90975
\(248\) 2.86563e8 1.19300
\(249\) 0 0
\(250\) 3.38444e8 1.36993
\(251\) 3.50809e8 1.40027 0.700137 0.714009i \(-0.253122\pi\)
0.700137 + 0.714009i \(0.253122\pi\)
\(252\) 0 0
\(253\) −2.28093e8 −0.885505
\(254\) −2.40736e8 −0.921772
\(255\) 0 0
\(256\) −3.78416e8 −1.40971
\(257\) 3.34359e8 1.22871 0.614353 0.789032i \(-0.289417\pi\)
0.614353 + 0.789032i \(0.289417\pi\)
\(258\) 0 0
\(259\) −7.57886e8 −2.71053
\(260\) 1.08148e9 3.81601
\(261\) 0 0
\(262\) −1.92855e8 −0.662485
\(263\) 3.99920e8 1.35559 0.677795 0.735251i \(-0.262936\pi\)
0.677795 + 0.735251i \(0.262936\pi\)
\(264\) 0 0
\(265\) −3.42324e8 −1.13000
\(266\) −1.31039e9 −4.26889
\(267\) 0 0
\(268\) 3.29407e8 1.04535
\(269\) −1.62396e7 −0.0508677 −0.0254338 0.999677i \(-0.508097\pi\)
−0.0254338 + 0.999677i \(0.508097\pi\)
\(270\) 0 0
\(271\) −4.27140e8 −1.30370 −0.651851 0.758347i \(-0.726007\pi\)
−0.651851 + 0.758347i \(0.726007\pi\)
\(272\) 7.91470e7 0.238475
\(273\) 0 0
\(274\) −8.85419e8 −2.60029
\(275\) 3.28485e8 0.952470
\(276\) 0 0
\(277\) −4.70787e8 −1.33090 −0.665449 0.746443i \(-0.731760\pi\)
−0.665449 + 0.746443i \(0.731760\pi\)
\(278\) −1.07940e9 −3.01319
\(279\) 0 0
\(280\) −1.32995e9 −3.62061
\(281\) 3.44673e8 0.926692 0.463346 0.886177i \(-0.346649\pi\)
0.463346 + 0.886177i \(0.346649\pi\)
\(282\) 0 0
\(283\) 3.05838e8 0.802120 0.401060 0.916052i \(-0.368642\pi\)
0.401060 + 0.916052i \(0.368642\pi\)
\(284\) −7.56902e8 −1.96076
\(285\) 0 0
\(286\) 5.66789e8 1.43265
\(287\) 6.29965e8 1.57300
\(288\) 0 0
\(289\) −1.15582e8 −0.281674
\(290\) −8.56473e8 −2.06215
\(291\) 0 0
\(292\) −8.75007e8 −2.05670
\(293\) 8.35017e7 0.193936 0.0969680 0.995287i \(-0.469086\pi\)
0.0969680 + 0.995287i \(0.469086\pi\)
\(294\) 0 0
\(295\) 9.11538e7 0.206727
\(296\) −7.89728e8 −1.76993
\(297\) 0 0
\(298\) 1.76995e8 0.387440
\(299\) 9.04286e8 1.95640
\(300\) 0 0
\(301\) 1.34035e9 2.83293
\(302\) 2.89343e8 0.604489
\(303\) 0 0
\(304\) −1.90305e8 −0.388502
\(305\) 9.40092e7 0.189723
\(306\) 0 0
\(307\) −7.34193e8 −1.44819 −0.724096 0.689699i \(-0.757743\pi\)
−0.724096 + 0.689699i \(0.757743\pi\)
\(308\) −1.04226e9 −2.03259
\(309\) 0 0
\(310\) −1.34733e9 −2.56867
\(311\) −2.34224e7 −0.0441540 −0.0220770 0.999756i \(-0.507028\pi\)
−0.0220770 + 0.999756i \(0.507028\pi\)
\(312\) 0 0
\(313\) 4.22581e8 0.778941 0.389471 0.921039i \(-0.372658\pi\)
0.389471 + 0.921039i \(0.372658\pi\)
\(314\) −6.78799e8 −1.23734
\(315\) 0 0
\(316\) 1.01277e9 1.80553
\(317\) 1.79803e7 0.0317023 0.0158511 0.999874i \(-0.494954\pi\)
0.0158511 + 0.999874i \(0.494954\pi\)
\(318\) 0 0
\(319\) −2.84897e8 −0.491383
\(320\) 1.42408e9 2.42945
\(321\) 0 0
\(322\) −2.61994e9 −4.37317
\(323\) −7.08729e8 −1.17023
\(324\) 0 0
\(325\) −1.30229e9 −2.10435
\(326\) −9.87695e8 −1.57892
\(327\) 0 0
\(328\) 6.56432e8 1.02714
\(329\) −1.88251e9 −2.91442
\(330\) 0 0
\(331\) −2.19639e8 −0.332899 −0.166449 0.986050i \(-0.553230\pi\)
−0.166449 + 0.986050i \(0.553230\pi\)
\(332\) 1.00514e9 1.50745
\(333\) 0 0
\(334\) −1.37893e8 −0.202502
\(335\) −6.57386e8 −0.955352
\(336\) 0 0
\(337\) 4.32909e7 0.0616158 0.0308079 0.999525i \(-0.490192\pi\)
0.0308079 + 0.999525i \(0.490192\pi\)
\(338\) −1.07247e9 −1.51070
\(339\) 0 0
\(340\) −1.69466e9 −2.33833
\(341\) −4.48176e8 −0.612080
\(342\) 0 0
\(343\) −2.08346e9 −2.78777
\(344\) 1.39667e9 1.84986
\(345\) 0 0
\(346\) 1.47419e9 1.91332
\(347\) −2.07119e8 −0.266114 −0.133057 0.991108i \(-0.542479\pi\)
−0.133057 + 0.991108i \(0.542479\pi\)
\(348\) 0 0
\(349\) −1.35268e9 −1.70336 −0.851680 0.524062i \(-0.824416\pi\)
−0.851680 + 0.524062i \(0.824416\pi\)
\(350\) 3.77307e9 4.70389
\(351\) 0 0
\(352\) 3.86587e8 0.472442
\(353\) 9.24750e8 1.11896 0.559478 0.828845i \(-0.311002\pi\)
0.559478 + 0.828845i \(0.311002\pi\)
\(354\) 0 0
\(355\) 1.51052e9 1.79196
\(356\) 9.37560e8 1.10135
\(357\) 0 0
\(358\) 3.99486e8 0.460162
\(359\) −5.95223e8 −0.678968 −0.339484 0.940612i \(-0.610252\pi\)
−0.339484 + 0.940612i \(0.610252\pi\)
\(360\) 0 0
\(361\) 8.10232e8 0.906430
\(362\) −3.22421e8 −0.357226
\(363\) 0 0
\(364\) 4.13209e9 4.49071
\(365\) 1.74622e9 1.87964
\(366\) 0 0
\(367\) 5.60873e8 0.592289 0.296144 0.955143i \(-0.404299\pi\)
0.296144 + 0.955143i \(0.404299\pi\)
\(368\) −3.80488e8 −0.397992
\(369\) 0 0
\(370\) 3.71306e9 3.81089
\(371\) −1.30795e9 −1.32979
\(372\) 0 0
\(373\) −6.85459e8 −0.683912 −0.341956 0.939716i \(-0.611089\pi\)
−0.341956 + 0.939716i \(0.611089\pi\)
\(374\) −8.88150e8 −0.877881
\(375\) 0 0
\(376\) −1.96160e9 −1.90306
\(377\) 1.12949e9 1.08564
\(378\) 0 0
\(379\) −1.03369e9 −0.975337 −0.487668 0.873029i \(-0.662152\pi\)
−0.487668 + 0.873029i \(0.662152\pi\)
\(380\) 4.07472e9 3.80939
\(381\) 0 0
\(382\) 4.98632e8 0.457676
\(383\) −2.09792e9 −1.90806 −0.954032 0.299705i \(-0.903112\pi\)
−0.954032 + 0.299705i \(0.903112\pi\)
\(384\) 0 0
\(385\) 2.08000e9 1.85760
\(386\) 3.17582e9 2.81061
\(387\) 0 0
\(388\) −3.55670e8 −0.309126
\(389\) 6.60600e7 0.0569003 0.0284502 0.999595i \(-0.490943\pi\)
0.0284502 + 0.999595i \(0.490943\pi\)
\(390\) 0 0
\(391\) −1.41700e9 −1.19882
\(392\) −3.62623e9 −3.04056
\(393\) 0 0
\(394\) 1.20069e9 0.988998
\(395\) −2.02115e9 −1.65009
\(396\) 0 0
\(397\) 1.83981e9 1.47573 0.737863 0.674951i \(-0.235835\pi\)
0.737863 + 0.674951i \(0.235835\pi\)
\(398\) 2.14868e9 1.70837
\(399\) 0 0
\(400\) 5.47955e8 0.428090
\(401\) −1.61549e9 −1.25112 −0.625560 0.780176i \(-0.715130\pi\)
−0.625560 + 0.780176i \(0.715130\pi\)
\(402\) 0 0
\(403\) 1.77681e9 1.35230
\(404\) 3.08446e8 0.232726
\(405\) 0 0
\(406\) −3.27240e9 −2.42675
\(407\) 1.23511e9 0.908083
\(408\) 0 0
\(409\) −1.07871e9 −0.779605 −0.389802 0.920899i \(-0.627457\pi\)
−0.389802 + 0.920899i \(0.627457\pi\)
\(410\) −3.08634e9 −2.21157
\(411\) 0 0
\(412\) −2.47311e8 −0.174222
\(413\) 3.48279e8 0.243278
\(414\) 0 0
\(415\) −2.00591e9 −1.37767
\(416\) −1.53264e9 −1.04379
\(417\) 0 0
\(418\) 2.13551e9 1.43016
\(419\) 8.10783e8 0.538463 0.269231 0.963076i \(-0.413230\pi\)
0.269231 + 0.963076i \(0.413230\pi\)
\(420\) 0 0
\(421\) −1.13878e9 −0.743792 −0.371896 0.928274i \(-0.621292\pi\)
−0.371896 + 0.928274i \(0.621292\pi\)
\(422\) −1.27192e8 −0.0823886
\(423\) 0 0
\(424\) −1.36290e9 −0.868327
\(425\) 2.04068e9 1.28948
\(426\) 0 0
\(427\) 3.59189e8 0.223268
\(428\) 5.34720e9 3.29665
\(429\) 0 0
\(430\) −6.56670e9 −3.98297
\(431\) 2.09869e9 1.26263 0.631317 0.775525i \(-0.282515\pi\)
0.631317 + 0.775525i \(0.282515\pi\)
\(432\) 0 0
\(433\) 1.85715e9 1.09936 0.549679 0.835376i \(-0.314750\pi\)
0.549679 + 0.835376i \(0.314750\pi\)
\(434\) −5.14787e9 −3.02283
\(435\) 0 0
\(436\) 2.36366e9 1.36579
\(437\) 3.40712e9 1.95300
\(438\) 0 0
\(439\) 6.58145e8 0.371275 0.185638 0.982618i \(-0.440565\pi\)
0.185638 + 0.982618i \(0.440565\pi\)
\(440\) 2.16739e9 1.21298
\(441\) 0 0
\(442\) 3.52111e9 1.93955
\(443\) 2.56332e8 0.140084 0.0700422 0.997544i \(-0.477687\pi\)
0.0700422 + 0.997544i \(0.477687\pi\)
\(444\) 0 0
\(445\) −1.87105e9 −1.00653
\(446\) −5.22056e9 −2.78641
\(447\) 0 0
\(448\) 5.44110e9 2.85900
\(449\) −5.70165e8 −0.297261 −0.148631 0.988893i \(-0.547487\pi\)
−0.148631 + 0.988893i \(0.547487\pi\)
\(450\) 0 0
\(451\) −1.02664e9 −0.526987
\(452\) −4.05575e9 −2.06579
\(453\) 0 0
\(454\) 7.86086e8 0.394253
\(455\) −8.24626e9 −4.10409
\(456\) 0 0
\(457\) −1.42553e8 −0.0698669 −0.0349334 0.999390i \(-0.511122\pi\)
−0.0349334 + 0.999390i \(0.511122\pi\)
\(458\) 2.82208e8 0.137259
\(459\) 0 0
\(460\) 8.14684e9 3.90245
\(461\) −2.44186e9 −1.16083 −0.580414 0.814321i \(-0.697109\pi\)
−0.580414 + 0.814321i \(0.697109\pi\)
\(462\) 0 0
\(463\) −2.57981e9 −1.20796 −0.603982 0.796998i \(-0.706420\pi\)
−0.603982 + 0.796998i \(0.706420\pi\)
\(464\) −4.75244e8 −0.220853
\(465\) 0 0
\(466\) 9.82495e8 0.449759
\(467\) 1.03284e9 0.469272 0.234636 0.972083i \(-0.424610\pi\)
0.234636 + 0.972083i \(0.424610\pi\)
\(468\) 0 0
\(469\) −2.51173e9 −1.12426
\(470\) 9.22285e9 4.09753
\(471\) 0 0
\(472\) 3.62912e8 0.158856
\(473\) −2.18434e9 −0.949089
\(474\) 0 0
\(475\) −4.90671e9 −2.10069
\(476\) −6.47493e9 −2.75176
\(477\) 0 0
\(478\) 3.41181e9 1.42885
\(479\) 2.54815e9 1.05938 0.529688 0.848193i \(-0.322309\pi\)
0.529688 + 0.848193i \(0.322309\pi\)
\(480\) 0 0
\(481\) −4.89665e9 −2.00628
\(482\) 1.03187e9 0.419719
\(483\) 0 0
\(484\) −2.63536e9 −1.05653
\(485\) 7.09797e8 0.282513
\(486\) 0 0
\(487\) −1.38050e9 −0.541608 −0.270804 0.962635i \(-0.587289\pi\)
−0.270804 + 0.962635i \(0.587289\pi\)
\(488\) 3.74280e8 0.145790
\(489\) 0 0
\(490\) 1.70494e10 6.54671
\(491\) −3.40284e9 −1.29735 −0.648674 0.761067i \(-0.724676\pi\)
−0.648674 + 0.761067i \(0.724676\pi\)
\(492\) 0 0
\(493\) −1.76989e9 −0.665246
\(494\) −8.46634e9 −3.15974
\(495\) 0 0
\(496\) −7.47614e8 −0.275101
\(497\) 5.77138e9 2.10879
\(498\) 0 0
\(499\) −2.71570e9 −0.978430 −0.489215 0.872163i \(-0.662717\pi\)
−0.489215 + 0.872163i \(0.662717\pi\)
\(500\) −4.02103e9 −1.43861
\(501\) 0 0
\(502\) −6.56677e9 −2.31680
\(503\) 1.12999e9 0.395899 0.197950 0.980212i \(-0.436572\pi\)
0.197950 + 0.980212i \(0.436572\pi\)
\(504\) 0 0
\(505\) −6.15554e8 −0.212690
\(506\) 4.26966e9 1.46510
\(507\) 0 0
\(508\) 2.86017e9 0.967986
\(509\) −2.58498e9 −0.868849 −0.434425 0.900708i \(-0.643048\pi\)
−0.434425 + 0.900708i \(0.643048\pi\)
\(510\) 0 0
\(511\) 6.67193e9 2.21197
\(512\) 1.68757e9 0.555670
\(513\) 0 0
\(514\) −6.25885e9 −2.03294
\(515\) 4.93550e8 0.159223
\(516\) 0 0
\(517\) 3.06788e9 0.976387
\(518\) 1.41868e10 4.48467
\(519\) 0 0
\(520\) −8.59272e9 −2.67990
\(521\) 1.68355e9 0.521546 0.260773 0.965400i \(-0.416023\pi\)
0.260773 + 0.965400i \(0.416023\pi\)
\(522\) 0 0
\(523\) −2.47225e9 −0.755676 −0.377838 0.925872i \(-0.623332\pi\)
−0.377838 + 0.925872i \(0.623332\pi\)
\(524\) 2.29129e9 0.695699
\(525\) 0 0
\(526\) −7.48608e9 −2.24287
\(527\) −2.78424e9 −0.828648
\(528\) 0 0
\(529\) 3.40723e9 1.00071
\(530\) 6.40794e9 1.86962
\(531\) 0 0
\(532\) 1.55687e10 4.48291
\(533\) 4.07016e9 1.16430
\(534\) 0 0
\(535\) −1.06712e10 −3.01284
\(536\) −2.61726e9 −0.734125
\(537\) 0 0
\(538\) 3.03988e8 0.0841624
\(539\) 5.67131e9 1.55999
\(540\) 0 0
\(541\) −3.60201e9 −0.978034 −0.489017 0.872274i \(-0.662644\pi\)
−0.489017 + 0.872274i \(0.662644\pi\)
\(542\) 7.99561e9 2.15702
\(543\) 0 0
\(544\) 2.40163e9 0.639602
\(545\) −4.71707e9 −1.24820
\(546\) 0 0
\(547\) 1.24077e9 0.324141 0.162071 0.986779i \(-0.448183\pi\)
0.162071 + 0.986779i \(0.448183\pi\)
\(548\) 1.05196e10 2.73066
\(549\) 0 0
\(550\) −6.14889e9 −1.57590
\(551\) 4.25561e9 1.08376
\(552\) 0 0
\(553\) −7.72237e9 −1.94184
\(554\) 8.81262e9 2.20202
\(555\) 0 0
\(556\) 1.28243e10 3.16426
\(557\) −1.77982e9 −0.436399 −0.218200 0.975904i \(-0.570018\pi\)
−0.218200 + 0.975904i \(0.570018\pi\)
\(558\) 0 0
\(559\) 8.65993e9 2.09688
\(560\) 3.46970e9 0.834900
\(561\) 0 0
\(562\) −6.45192e9 −1.53325
\(563\) 6.60782e9 1.56056 0.780278 0.625433i \(-0.215078\pi\)
0.780278 + 0.625433i \(0.215078\pi\)
\(564\) 0 0
\(565\) 8.09391e9 1.88794
\(566\) −5.72496e9 −1.32714
\(567\) 0 0
\(568\) 6.01386e9 1.37700
\(569\) −4.59260e9 −1.04512 −0.522560 0.852603i \(-0.675023\pi\)
−0.522560 + 0.852603i \(0.675023\pi\)
\(570\) 0 0
\(571\) 3.76881e9 0.847183 0.423592 0.905853i \(-0.360769\pi\)
0.423592 + 0.905853i \(0.360769\pi\)
\(572\) −6.73398e9 −1.50448
\(573\) 0 0
\(574\) −1.17923e10 −2.60259
\(575\) −9.81029e9 −2.15201
\(576\) 0 0
\(577\) 3.58209e9 0.776284 0.388142 0.921600i \(-0.373117\pi\)
0.388142 + 0.921600i \(0.373117\pi\)
\(578\) 2.16357e9 0.466040
\(579\) 0 0
\(580\) 1.01757e10 2.16554
\(581\) −7.66417e9 −1.62125
\(582\) 0 0
\(583\) 2.13153e9 0.445504
\(584\) 6.95225e9 1.44438
\(585\) 0 0
\(586\) −1.56306e9 −0.320874
\(587\) −3.85835e8 −0.0787350 −0.0393675 0.999225i \(-0.512534\pi\)
−0.0393675 + 0.999225i \(0.512534\pi\)
\(588\) 0 0
\(589\) 6.69458e9 1.34996
\(590\) −1.70630e9 −0.342037
\(591\) 0 0
\(592\) 2.06032e9 0.408140
\(593\) 1.84467e9 0.363267 0.181634 0.983366i \(-0.441861\pi\)
0.181634 + 0.983366i \(0.441861\pi\)
\(594\) 0 0
\(595\) 1.29218e10 2.51485
\(596\) −2.10287e9 −0.406864
\(597\) 0 0
\(598\) −1.69273e10 −3.23693
\(599\) −4.66546e9 −0.886953 −0.443476 0.896286i \(-0.646255\pi\)
−0.443476 + 0.896286i \(0.646255\pi\)
\(600\) 0 0
\(601\) 2.24604e9 0.422043 0.211021 0.977481i \(-0.432321\pi\)
0.211021 + 0.977481i \(0.432321\pi\)
\(602\) −2.50900e10 −4.68719
\(603\) 0 0
\(604\) −3.43766e9 −0.634796
\(605\) 5.25930e9 0.965570
\(606\) 0 0
\(607\) 9.43363e9 1.71206 0.856029 0.516928i \(-0.172925\pi\)
0.856029 + 0.516928i \(0.172925\pi\)
\(608\) −5.77460e9 −1.04198
\(609\) 0 0
\(610\) −1.75975e9 −0.313904
\(611\) −1.21628e10 −2.15719
\(612\) 0 0
\(613\) 2.56081e9 0.449020 0.224510 0.974472i \(-0.427922\pi\)
0.224510 + 0.974472i \(0.427922\pi\)
\(614\) 1.37433e10 2.39608
\(615\) 0 0
\(616\) 8.28114e9 1.42744
\(617\) −2.42876e9 −0.416281 −0.208141 0.978099i \(-0.566741\pi\)
−0.208141 + 0.978099i \(0.566741\pi\)
\(618\) 0 0
\(619\) 6.53273e9 1.10708 0.553538 0.832824i \(-0.313277\pi\)
0.553538 + 0.832824i \(0.313277\pi\)
\(620\) 1.60076e10 2.69746
\(621\) 0 0
\(622\) 4.38443e8 0.0730544
\(623\) −7.14890e9 −1.18449
\(624\) 0 0
\(625\) −1.26146e9 −0.206678
\(626\) −7.91026e9 −1.28879
\(627\) 0 0
\(628\) 8.06476e9 1.29937
\(629\) 7.67299e9 1.22938
\(630\) 0 0
\(631\) −2.21286e9 −0.350631 −0.175315 0.984512i \(-0.556095\pi\)
−0.175315 + 0.984512i \(0.556095\pi\)
\(632\) −8.04681e9 −1.26799
\(633\) 0 0
\(634\) −3.36573e8 −0.0524525
\(635\) −5.70794e9 −0.884649
\(636\) 0 0
\(637\) −2.24842e10 −3.44658
\(638\) 5.33296e9 0.813011
\(639\) 0 0
\(640\) −1.87102e10 −2.82130
\(641\) −5.20588e9 −0.780713 −0.390356 0.920664i \(-0.627648\pi\)
−0.390356 + 0.920664i \(0.627648\pi\)
\(642\) 0 0
\(643\) −1.18888e10 −1.76359 −0.881796 0.471631i \(-0.843665\pi\)
−0.881796 + 0.471631i \(0.843665\pi\)
\(644\) 3.11274e10 4.59242
\(645\) 0 0
\(646\) 1.32666e10 1.93619
\(647\) 6.02914e9 0.875166 0.437583 0.899178i \(-0.355835\pi\)
0.437583 + 0.899178i \(0.355835\pi\)
\(648\) 0 0
\(649\) −5.67583e8 −0.0815029
\(650\) 2.43776e10 3.48172
\(651\) 0 0
\(652\) 1.17347e10 1.65809
\(653\) −2.77068e8 −0.0389395 −0.0194697 0.999810i \(-0.506198\pi\)
−0.0194697 + 0.999810i \(0.506198\pi\)
\(654\) 0 0
\(655\) −4.57265e9 −0.635804
\(656\) −1.71256e9 −0.236856
\(657\) 0 0
\(658\) 3.52386e10 4.82200
\(659\) 1.19887e10 1.63182 0.815909 0.578181i \(-0.196237\pi\)
0.815909 + 0.578181i \(0.196237\pi\)
\(660\) 0 0
\(661\) 3.39523e9 0.457261 0.228631 0.973513i \(-0.426575\pi\)
0.228631 + 0.973513i \(0.426575\pi\)
\(662\) 4.11142e9 0.550793
\(663\) 0 0
\(664\) −7.98617e9 −1.05865
\(665\) −3.10698e10 −4.09697
\(666\) 0 0
\(667\) 8.50851e9 1.11023
\(668\) 1.63829e9 0.212654
\(669\) 0 0
\(670\) 1.23056e10 1.58066
\(671\) −5.85363e8 −0.0747991
\(672\) 0 0
\(673\) 9.44306e9 1.19415 0.597077 0.802184i \(-0.296329\pi\)
0.597077 + 0.802184i \(0.296329\pi\)
\(674\) −8.10359e8 −0.101945
\(675\) 0 0
\(676\) 1.27420e10 1.58644
\(677\) −2.31127e9 −0.286279 −0.143140 0.989703i \(-0.545720\pi\)
−0.143140 + 0.989703i \(0.545720\pi\)
\(678\) 0 0
\(679\) 2.71199e9 0.332463
\(680\) 1.34647e10 1.64216
\(681\) 0 0
\(682\) 8.38938e9 1.01271
\(683\) 1.31138e10 1.57491 0.787454 0.616374i \(-0.211399\pi\)
0.787454 + 0.616374i \(0.211399\pi\)
\(684\) 0 0
\(685\) −2.09936e10 −2.49557
\(686\) 3.90002e10 4.61246
\(687\) 0 0
\(688\) −3.64376e9 −0.426570
\(689\) −8.45056e9 −0.984279
\(690\) 0 0
\(691\) 8.33408e9 0.960913 0.480457 0.877018i \(-0.340471\pi\)
0.480457 + 0.877018i \(0.340471\pi\)
\(692\) −1.75148e10 −2.00925
\(693\) 0 0
\(694\) 3.87705e9 0.440295
\(695\) −2.55930e10 −2.89184
\(696\) 0 0
\(697\) −6.37789e9 −0.713448
\(698\) 2.53207e10 2.81827
\(699\) 0 0
\(700\) −4.48276e10 −4.93972
\(701\) 2.09612e9 0.229829 0.114914 0.993375i \(-0.463341\pi\)
0.114914 + 0.993375i \(0.463341\pi\)
\(702\) 0 0
\(703\) −1.84493e10 −2.00280
\(704\) −8.86724e9 −0.957821
\(705\) 0 0
\(706\) −1.73103e10 −1.85135
\(707\) −2.35190e9 −0.250294
\(708\) 0 0
\(709\) 4.17621e8 0.0440069 0.0220035 0.999758i \(-0.492996\pi\)
0.0220035 + 0.999758i \(0.492996\pi\)
\(710\) −2.82753e10 −2.96486
\(711\) 0 0
\(712\) −7.44925e9 −0.773451
\(713\) 1.33849e10 1.38293
\(714\) 0 0
\(715\) 1.34387e10 1.37495
\(716\) −4.74626e9 −0.483232
\(717\) 0 0
\(718\) 1.11419e10 1.12338
\(719\) 6.71535e9 0.673779 0.336890 0.941544i \(-0.390625\pi\)
0.336890 + 0.941544i \(0.390625\pi\)
\(720\) 0 0
\(721\) 1.88575e9 0.187375
\(722\) −1.51667e10 −1.49972
\(723\) 0 0
\(724\) 3.83066e9 0.375136
\(725\) −1.22534e10 −1.19419
\(726\) 0 0
\(727\) −8.79100e9 −0.848531 −0.424266 0.905538i \(-0.639468\pi\)
−0.424266 + 0.905538i \(0.639468\pi\)
\(728\) −3.28309e10 −3.15372
\(729\) 0 0
\(730\) −3.26874e10 −3.10992
\(731\) −1.35700e10 −1.28490
\(732\) 0 0
\(733\) −1.87568e10 −1.75912 −0.879560 0.475787i \(-0.842163\pi\)
−0.879560 + 0.475787i \(0.842163\pi\)
\(734\) −1.04989e10 −0.979963
\(735\) 0 0
\(736\) −1.15455e10 −1.06743
\(737\) 4.09331e9 0.376651
\(738\) 0 0
\(739\) −3.83197e9 −0.349274 −0.174637 0.984633i \(-0.555875\pi\)
−0.174637 + 0.984633i \(0.555875\pi\)
\(740\) −4.41146e10 −4.00195
\(741\) 0 0
\(742\) 2.44834e10 2.20018
\(743\) 1.74075e9 0.155695 0.0778475 0.996965i \(-0.475195\pi\)
0.0778475 + 0.996965i \(0.475195\pi\)
\(744\) 0 0
\(745\) 4.19661e9 0.371836
\(746\) 1.28311e10 1.13156
\(747\) 0 0
\(748\) 1.05521e10 0.921895
\(749\) −4.07724e10 −3.54552
\(750\) 0 0
\(751\) 1.87554e10 1.61580 0.807900 0.589320i \(-0.200604\pi\)
0.807900 + 0.589320i \(0.200604\pi\)
\(752\) 5.11762e9 0.438839
\(753\) 0 0
\(754\) −2.11428e10 −1.79623
\(755\) 6.86042e9 0.580145
\(756\) 0 0
\(757\) −1.48636e10 −1.24534 −0.622671 0.782483i \(-0.713953\pi\)
−0.622671 + 0.782483i \(0.713953\pi\)
\(758\) 1.93496e10 1.61373
\(759\) 0 0
\(760\) −3.23751e10 −2.67525
\(761\) −2.55852e9 −0.210446 −0.105223 0.994449i \(-0.533556\pi\)
−0.105223 + 0.994449i \(0.533556\pi\)
\(762\) 0 0
\(763\) −1.80229e10 −1.46889
\(764\) −5.92421e9 −0.480622
\(765\) 0 0
\(766\) 3.92708e10 3.15696
\(767\) 2.25021e9 0.180069
\(768\) 0 0
\(769\) 1.98268e10 1.57221 0.786104 0.618094i \(-0.212095\pi\)
0.786104 + 0.618094i \(0.212095\pi\)
\(770\) −3.89354e10 −3.07346
\(771\) 0 0
\(772\) −3.77317e10 −2.95152
\(773\) 1.29417e10 1.00777 0.503886 0.863770i \(-0.331903\pi\)
0.503886 + 0.863770i \(0.331903\pi\)
\(774\) 0 0
\(775\) −1.92760e10 −1.48752
\(776\) 2.82593e9 0.217092
\(777\) 0 0
\(778\) −1.23657e9 −0.0941436
\(779\) 1.53353e10 1.16228
\(780\) 0 0
\(781\) −9.40549e9 −0.706485
\(782\) 2.65248e10 1.98348
\(783\) 0 0
\(784\) 9.46046e9 0.701142
\(785\) −1.60945e10 −1.18750
\(786\) 0 0
\(787\) 9.60328e9 0.702276 0.351138 0.936324i \(-0.385795\pi\)
0.351138 + 0.936324i \(0.385795\pi\)
\(788\) −1.42654e10 −1.03858
\(789\) 0 0
\(790\) 3.78337e10 2.73013
\(791\) 3.09251e10 2.22174
\(792\) 0 0
\(793\) 2.32070e9 0.165258
\(794\) −3.44392e10 −2.44164
\(795\) 0 0
\(796\) −2.55284e10 −1.79402
\(797\) 9.10764e9 0.637238 0.318619 0.947883i \(-0.396781\pi\)
0.318619 + 0.947883i \(0.396781\pi\)
\(798\) 0 0
\(799\) 1.90589e10 1.32185
\(800\) 1.66271e10 1.14816
\(801\) 0 0
\(802\) 3.02403e10 2.07002
\(803\) −1.08731e10 −0.741053
\(804\) 0 0
\(805\) −6.21197e10 −4.19705
\(806\) −3.32601e10 −2.23744
\(807\) 0 0
\(808\) −2.45071e9 −0.163438
\(809\) 1.78494e10 1.18524 0.592618 0.805484i \(-0.298094\pi\)
0.592618 + 0.805484i \(0.298094\pi\)
\(810\) 0 0
\(811\) 6.82320e9 0.449174 0.224587 0.974454i \(-0.427897\pi\)
0.224587 + 0.974454i \(0.427897\pi\)
\(812\) 3.88792e10 2.54842
\(813\) 0 0
\(814\) −2.31200e10 −1.50246
\(815\) −2.34186e10 −1.51534
\(816\) 0 0
\(817\) 3.26284e10 2.09324
\(818\) 2.01924e10 1.28988
\(819\) 0 0
\(820\) 3.66686e10 2.32245
\(821\) −7.07763e9 −0.446362 −0.223181 0.974777i \(-0.571644\pi\)
−0.223181 + 0.974777i \(0.571644\pi\)
\(822\) 0 0
\(823\) −1.02079e10 −0.638317 −0.319159 0.947701i \(-0.603400\pi\)
−0.319159 + 0.947701i \(0.603400\pi\)
\(824\) 1.96498e9 0.122352
\(825\) 0 0
\(826\) −6.51941e9 −0.402512
\(827\) −2.59808e9 −0.159729 −0.0798644 0.996806i \(-0.525449\pi\)
−0.0798644 + 0.996806i \(0.525449\pi\)
\(828\) 0 0
\(829\) 2.93686e9 0.179037 0.0895184 0.995985i \(-0.471467\pi\)
0.0895184 + 0.995985i \(0.471467\pi\)
\(830\) 3.75486e10 2.27940
\(831\) 0 0
\(832\) 3.51546e10 2.11617
\(833\) 3.52324e10 2.11195
\(834\) 0 0
\(835\) −3.26948e9 −0.194346
\(836\) −2.53719e10 −1.50187
\(837\) 0 0
\(838\) −1.51770e10 −0.890905
\(839\) −1.16450e10 −0.680728 −0.340364 0.940294i \(-0.610550\pi\)
−0.340364 + 0.940294i \(0.610550\pi\)
\(840\) 0 0
\(841\) −6.62244e9 −0.383912
\(842\) 2.13167e10 1.23063
\(843\) 0 0
\(844\) 1.51116e9 0.0865192
\(845\) −2.54287e10 −1.44986
\(846\) 0 0
\(847\) 2.00947e10 1.13629
\(848\) 3.55567e9 0.200233
\(849\) 0 0
\(850\) −3.81993e10 −2.13348
\(851\) −3.68869e10 −2.05172
\(852\) 0 0
\(853\) 1.63671e10 0.902922 0.451461 0.892291i \(-0.350903\pi\)
0.451461 + 0.892291i \(0.350903\pi\)
\(854\) −6.72364e9 −0.369404
\(855\) 0 0
\(856\) −4.24854e10 −2.31517
\(857\) 1.66058e10 0.901210 0.450605 0.892724i \(-0.351208\pi\)
0.450605 + 0.892724i \(0.351208\pi\)
\(858\) 0 0
\(859\) −1.88831e10 −1.01647 −0.508237 0.861217i \(-0.669703\pi\)
−0.508237 + 0.861217i \(0.669703\pi\)
\(860\) 7.80185e10 4.18266
\(861\) 0 0
\(862\) −3.92852e10 −2.08907
\(863\) −2.09759e10 −1.11092 −0.555461 0.831543i \(-0.687458\pi\)
−0.555461 + 0.831543i \(0.687458\pi\)
\(864\) 0 0
\(865\) 3.49536e10 1.83627
\(866\) −3.47639e10 −1.81893
\(867\) 0 0
\(868\) 6.11615e10 3.17438
\(869\) 1.25850e10 0.650554
\(870\) 0 0
\(871\) −1.62281e10 −0.832156
\(872\) −1.87801e10 −0.959161
\(873\) 0 0
\(874\) −6.37776e10 −3.23131
\(875\) 3.06604e10 1.54721
\(876\) 0 0
\(877\) 3.68896e10 1.84674 0.923370 0.383911i \(-0.125423\pi\)
0.923370 + 0.383911i \(0.125423\pi\)
\(878\) −1.23198e10 −0.614288
\(879\) 0 0
\(880\) −5.65450e9 −0.279708
\(881\) 3.57144e10 1.75966 0.879829 0.475291i \(-0.157657\pi\)
0.879829 + 0.475291i \(0.157657\pi\)
\(882\) 0 0
\(883\) −3.90952e9 −0.191100 −0.0955501 0.995425i \(-0.530461\pi\)
−0.0955501 + 0.995425i \(0.530461\pi\)
\(884\) −4.18341e10 −2.03679
\(885\) 0 0
\(886\) −4.79826e9 −0.231775
\(887\) 1.25223e10 0.602494 0.301247 0.953546i \(-0.402597\pi\)
0.301247 + 0.953546i \(0.402597\pi\)
\(888\) 0 0
\(889\) −2.18088e10 −1.04106
\(890\) 3.50241e10 1.66534
\(891\) 0 0
\(892\) 6.20251e10 2.92611
\(893\) −4.58262e10 −2.15344
\(894\) 0 0
\(895\) 9.47193e9 0.441630
\(896\) −7.14878e10 −3.32012
\(897\) 0 0
\(898\) 1.06729e10 0.491829
\(899\) 1.67182e10 0.767416
\(900\) 0 0
\(901\) 1.32419e10 0.603134
\(902\) 1.92176e10 0.871919
\(903\) 0 0
\(904\) 3.22244e10 1.45076
\(905\) −7.64470e9 −0.342839
\(906\) 0 0
\(907\) 2.27291e10 1.01148 0.505739 0.862687i \(-0.331220\pi\)
0.505739 + 0.862687i \(0.331220\pi\)
\(908\) −9.33944e9 −0.414019
\(909\) 0 0
\(910\) 1.54361e11 6.79036
\(911\) 2.77093e10 1.21426 0.607129 0.794603i \(-0.292321\pi\)
0.607129 + 0.794603i \(0.292321\pi\)
\(912\) 0 0
\(913\) 1.24901e10 0.543150
\(914\) 2.66845e9 0.115597
\(915\) 0 0
\(916\) −3.35289e9 −0.144140
\(917\) −1.74711e10 −0.748218
\(918\) 0 0
\(919\) −1.24377e10 −0.528612 −0.264306 0.964439i \(-0.585143\pi\)
−0.264306 + 0.964439i \(0.585143\pi\)
\(920\) −6.47296e10 −2.74060
\(921\) 0 0
\(922\) 4.57091e10 1.92063
\(923\) 3.72885e10 1.56088
\(924\) 0 0
\(925\) 5.31221e10 2.20688
\(926\) 4.82913e10 1.99862
\(927\) 0 0
\(928\) −1.44208e10 −0.592339
\(929\) 3.14739e10 1.28794 0.643970 0.765051i \(-0.277286\pi\)
0.643970 + 0.765051i \(0.277286\pi\)
\(930\) 0 0
\(931\) −8.47146e10 −3.44060
\(932\) −1.16730e10 −0.472308
\(933\) 0 0
\(934\) −1.93337e10 −0.776428
\(935\) −2.10583e10 −0.842526
\(936\) 0 0
\(937\) −3.85637e10 −1.53140 −0.765702 0.643195i \(-0.777608\pi\)
−0.765702 + 0.643195i \(0.777608\pi\)
\(938\) 4.70169e10 1.86013
\(939\) 0 0
\(940\) −1.09576e11 −4.30297
\(941\) −2.15748e10 −0.844078 −0.422039 0.906578i \(-0.638686\pi\)
−0.422039 + 0.906578i \(0.638686\pi\)
\(942\) 0 0
\(943\) 3.06608e10 1.19067
\(944\) −9.46800e8 −0.0366316
\(945\) 0 0
\(946\) 4.08886e10 1.57030
\(947\) 2.58130e10 0.987675 0.493838 0.869554i \(-0.335594\pi\)
0.493838 + 0.869554i \(0.335594\pi\)
\(948\) 0 0
\(949\) 4.31069e10 1.63725
\(950\) 9.18484e10 3.47567
\(951\) 0 0
\(952\) 5.14457e10 1.93250
\(953\) −3.92406e10 −1.46862 −0.734311 0.678813i \(-0.762495\pi\)
−0.734311 + 0.678813i \(0.762495\pi\)
\(954\) 0 0
\(955\) 1.18227e10 0.439244
\(956\) −4.05355e10 −1.50049
\(957\) 0 0
\(958\) −4.76986e10 −1.75278
\(959\) −8.02121e10 −2.93680
\(960\) 0 0
\(961\) −1.21292e9 −0.0440859
\(962\) 9.16601e10 3.31946
\(963\) 0 0
\(964\) −1.22595e10 −0.440762
\(965\) 7.52997e10 2.69741
\(966\) 0 0
\(967\) −1.78901e9 −0.0636240 −0.0318120 0.999494i \(-0.510128\pi\)
−0.0318120 + 0.999494i \(0.510128\pi\)
\(968\) 2.09389e10 0.741977
\(969\) 0 0
\(970\) −1.32866e10 −0.467427
\(971\) 2.44481e10 0.856993 0.428497 0.903543i \(-0.359043\pi\)
0.428497 + 0.903543i \(0.359043\pi\)
\(972\) 0 0
\(973\) −9.77854e10 −3.40313
\(974\) 2.58415e10 0.896109
\(975\) 0 0
\(976\) −9.76459e8 −0.0336186
\(977\) −3.42307e10 −1.17432 −0.587158 0.809472i \(-0.699753\pi\)
−0.587158 + 0.809472i \(0.699753\pi\)
\(978\) 0 0
\(979\) 1.16504e10 0.396827
\(980\) −2.02563e11 −6.87494
\(981\) 0 0
\(982\) 6.36975e10 2.14651
\(983\) 4.53628e10 1.52322 0.761610 0.648036i \(-0.224409\pi\)
0.761610 + 0.648036i \(0.224409\pi\)
\(984\) 0 0
\(985\) 2.84688e10 0.949167
\(986\) 3.31304e10 1.10067
\(987\) 0 0
\(988\) 1.00588e11 3.31816
\(989\) 6.52359e10 2.14437
\(990\) 0 0
\(991\) 2.84050e10 0.927123 0.463561 0.886065i \(-0.346571\pi\)
0.463561 + 0.886065i \(0.346571\pi\)
\(992\) −2.26855e10 −0.737834
\(993\) 0 0
\(994\) −1.08034e11 −3.48906
\(995\) 5.09460e10 1.63957
\(996\) 0 0
\(997\) −3.63738e10 −1.16240 −0.581201 0.813760i \(-0.697417\pi\)
−0.581201 + 0.813760i \(0.697417\pi\)
\(998\) 5.08350e10 1.61885
\(999\) 0 0
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 531.8.a.b.1.2 16
3.2 odd 2 177.8.a.a.1.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.15 16 3.2 odd 2
531.8.a.b.1.2 16 1.1 even 1 trivial