Properties

Label 531.8.a.e.1.6
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} - 774932111214 x^{9} - 14600047830166 x^{8} + 102185027148481 x^{7} + 988557475638619 x^{6} - 7379206238519716 x^{5} - 30152342836849520 x^{4} + 260578770749067175 x^{3} + 182609347488069978 x^{2} - 3481290425710753600 x + 5164646074739714048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(9.20800\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-10.2080 q^{2} -23.7967 q^{4} +397.678 q^{5} -505.577 q^{7} +1549.54 q^{8} +O(q^{10})\) \(q-10.2080 q^{2} -23.7967 q^{4} +397.678 q^{5} -505.577 q^{7} +1549.54 q^{8} -4059.50 q^{10} +5491.59 q^{11} +5262.48 q^{13} +5160.93 q^{14} -12771.7 q^{16} -33639.9 q^{17} +41812.3 q^{19} -9463.44 q^{20} -56058.1 q^{22} +14742.3 q^{23} +80022.8 q^{25} -53719.4 q^{26} +12031.1 q^{28} -192269. q^{29} -222873. q^{31} -67967.4 q^{32} +343396. q^{34} -201057. q^{35} +319803. q^{37} -426820. q^{38} +616218. q^{40} +210002. q^{41} -197848. q^{43} -130682. q^{44} -150490. q^{46} -1.24221e6 q^{47} -567935. q^{49} -816873. q^{50} -125230. q^{52} +346356. q^{53} +2.18388e6 q^{55} -783412. q^{56} +1.96269e6 q^{58} -205379. q^{59} -2.78811e6 q^{61} +2.27509e6 q^{62} +2.32859e6 q^{64} +2.09277e6 q^{65} -4.61431e6 q^{67} +800519. q^{68} +2.05239e6 q^{70} -1.49916e6 q^{71} -1.38946e6 q^{73} -3.26455e6 q^{74} -994997. q^{76} -2.77642e6 q^{77} +7.13579e6 q^{79} -5.07904e6 q^{80} -2.14370e6 q^{82} -2.03328e6 q^{83} -1.33778e7 q^{85} +2.01963e6 q^{86} +8.50944e6 q^{88} -5.52152e6 q^{89} -2.66059e6 q^{91} -350820. q^{92} +1.26805e7 q^{94} +1.66278e7 q^{95} +1.50671e7 q^{97} +5.79748e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 24q^{2} + 1358q^{4} - 678q^{5} + 3081q^{7} - 4107q^{8} + O(q^{10}) \) \( 18q - 24q^{2} + 1358q^{4} - 678q^{5} + 3081q^{7} - 4107q^{8} + 3609q^{10} - 15070q^{11} + 13662q^{13} - 20861q^{14} + 60482q^{16} - 71919q^{17} + 56231q^{19} - 143053q^{20} + 274198q^{22} - 150029q^{23} + 399672q^{25} - 182846q^{26} + 434150q^{28} - 591285q^{29} + 426733q^{31} - 1205630q^{32} + 403548q^{34} - 912879q^{35} + 7703q^{37} + 417859q^{38} + 618020q^{40} - 770959q^{41} + 793050q^{43} - 2591274q^{44} - 4068019q^{46} - 1410373q^{47} + 1637427q^{49} - 1021549q^{50} - 3749190q^{52} - 1037934q^{53} + 331974q^{55} + 391748q^{56} + 653724q^{58} - 3696822q^{59} - 1374623q^{61} - 5251718q^{62} + 5077197q^{64} - 3257170q^{65} - 2436904q^{67} - 14119909q^{68} + 5185580q^{70} - 14289172q^{71} + 5482515q^{73} - 14934154q^{74} + 3822912q^{76} - 23157109q^{77} + 19786414q^{79} - 31978143q^{80} + 9749509q^{82} - 30227337q^{83} + 9946981q^{85} - 44295864q^{86} + 39970897q^{88} - 31061677q^{89} + 26377785q^{91} - 4719698q^{92} + 44488296q^{94} - 15534599q^{95} + 12084118q^{97} - 42274744q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.2080 −0.902268 −0.451134 0.892456i \(-0.648980\pi\)
−0.451134 + 0.892456i \(0.648980\pi\)
\(3\) 0 0
\(4\) −23.7967 −0.185912
\(5\) 397.678 1.42278 0.711388 0.702799i \(-0.248067\pi\)
0.711388 + 0.702799i \(0.248067\pi\)
\(6\) 0 0
\(7\) −505.577 −0.557114 −0.278557 0.960420i \(-0.589856\pi\)
−0.278557 + 0.960420i \(0.589856\pi\)
\(8\) 1549.54 1.07001
\(9\) 0 0
\(10\) −4059.50 −1.28373
\(11\) 5491.59 1.24401 0.622004 0.783014i \(-0.286319\pi\)
0.622004 + 0.783014i \(0.286319\pi\)
\(12\) 0 0
\(13\) 5262.48 0.664338 0.332169 0.943220i \(-0.392220\pi\)
0.332169 + 0.943220i \(0.392220\pi\)
\(14\) 5160.93 0.502666
\(15\) 0 0
\(16\) −12771.7 −0.779525
\(17\) −33639.9 −1.66067 −0.830334 0.557265i \(-0.811851\pi\)
−0.830334 + 0.557265i \(0.811851\pi\)
\(18\) 0 0
\(19\) 41812.3 1.39851 0.699257 0.714871i \(-0.253514\pi\)
0.699257 + 0.714871i \(0.253514\pi\)
\(20\) −9463.44 −0.264511
\(21\) 0 0
\(22\) −56058.1 −1.12243
\(23\) 14742.3 0.252650 0.126325 0.991989i \(-0.459682\pi\)
0.126325 + 0.991989i \(0.459682\pi\)
\(24\) 0 0
\(25\) 80022.8 1.02429
\(26\) −53719.4 −0.599411
\(27\) 0 0
\(28\) 12031.1 0.103574
\(29\) −192269. −1.46392 −0.731960 0.681348i \(-0.761394\pi\)
−0.731960 + 0.681348i \(0.761394\pi\)
\(30\) 0 0
\(31\) −222873. −1.34367 −0.671833 0.740703i \(-0.734493\pi\)
−0.671833 + 0.740703i \(0.734493\pi\)
\(32\) −67967.4 −0.366670
\(33\) 0 0
\(34\) 343396. 1.49837
\(35\) −201057. −0.792648
\(36\) 0 0
\(37\) 319803. 1.03795 0.518975 0.854789i \(-0.326314\pi\)
0.518975 + 0.854789i \(0.326314\pi\)
\(38\) −426820. −1.26183
\(39\) 0 0
\(40\) 616218. 1.52239
\(41\) 210002. 0.475861 0.237930 0.971282i \(-0.423531\pi\)
0.237930 + 0.971282i \(0.423531\pi\)
\(42\) 0 0
\(43\) −197848. −0.379483 −0.189741 0.981834i \(-0.560765\pi\)
−0.189741 + 0.981834i \(0.560765\pi\)
\(44\) −130682. −0.231276
\(45\) 0 0
\(46\) −150490. −0.227958
\(47\) −1.24221e6 −1.74523 −0.872614 0.488410i \(-0.837577\pi\)
−0.872614 + 0.488410i \(0.837577\pi\)
\(48\) 0 0
\(49\) −567935. −0.689624
\(50\) −816873. −0.924186
\(51\) 0 0
\(52\) −125230. −0.123508
\(53\) 346356. 0.319564 0.159782 0.987152i \(-0.448921\pi\)
0.159782 + 0.987152i \(0.448921\pi\)
\(54\) 0 0
\(55\) 2.18388e6 1.76995
\(56\) −783412. −0.596118
\(57\) 0 0
\(58\) 1.96269e6 1.32085
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −2.78811e6 −1.57274 −0.786368 0.617758i \(-0.788041\pi\)
−0.786368 + 0.617758i \(0.788041\pi\)
\(62\) 2.27509e6 1.21235
\(63\) 0 0
\(64\) 2.32859e6 1.11036
\(65\) 2.09277e6 0.945204
\(66\) 0 0
\(67\) −4.61431e6 −1.87432 −0.937162 0.348895i \(-0.886557\pi\)
−0.937162 + 0.348895i \(0.886557\pi\)
\(68\) 800519. 0.308738
\(69\) 0 0
\(70\) 2.05239e6 0.715181
\(71\) −1.49916e6 −0.497101 −0.248550 0.968619i \(-0.579954\pi\)
−0.248550 + 0.968619i \(0.579954\pi\)
\(72\) 0 0
\(73\) −1.38946e6 −0.418039 −0.209019 0.977911i \(-0.567027\pi\)
−0.209019 + 0.977911i \(0.567027\pi\)
\(74\) −3.26455e6 −0.936510
\(75\) 0 0
\(76\) −994997. −0.260000
\(77\) −2.77642e6 −0.693055
\(78\) 0 0
\(79\) 7.13579e6 1.62835 0.814174 0.580621i \(-0.197190\pi\)
0.814174 + 0.580621i \(0.197190\pi\)
\(80\) −5.07904e6 −1.10909
\(81\) 0 0
\(82\) −2.14370e6 −0.429354
\(83\) −2.03328e6 −0.390322 −0.195161 0.980771i \(-0.562523\pi\)
−0.195161 + 0.980771i \(0.562523\pi\)
\(84\) 0 0
\(85\) −1.33778e7 −2.36276
\(86\) 2.01963e6 0.342395
\(87\) 0 0
\(88\) 8.50944e6 1.33110
\(89\) −5.52152e6 −0.830222 −0.415111 0.909771i \(-0.636257\pi\)
−0.415111 + 0.909771i \(0.636257\pi\)
\(90\) 0 0
\(91\) −2.66059e6 −0.370112
\(92\) −350820. −0.0469707
\(93\) 0 0
\(94\) 1.26805e7 1.57466
\(95\) 1.66278e7 1.98977
\(96\) 0 0
\(97\) 1.50671e7 1.67621 0.838103 0.545512i \(-0.183665\pi\)
0.838103 + 0.545512i \(0.183665\pi\)
\(98\) 5.79748e6 0.622226
\(99\) 0 0
\(100\) −1.90428e6 −0.190428
\(101\) 7.10580e6 0.686260 0.343130 0.939288i \(-0.388513\pi\)
0.343130 + 0.939288i \(0.388513\pi\)
\(102\) 0 0
\(103\) 5.41858e6 0.488602 0.244301 0.969699i \(-0.421442\pi\)
0.244301 + 0.969699i \(0.421442\pi\)
\(104\) 8.15444e6 0.710849
\(105\) 0 0
\(106\) −3.53560e6 −0.288332
\(107\) −1.98181e7 −1.56393 −0.781967 0.623320i \(-0.785784\pi\)
−0.781967 + 0.623320i \(0.785784\pi\)
\(108\) 0 0
\(109\) 9.80363e6 0.725094 0.362547 0.931966i \(-0.381907\pi\)
0.362547 + 0.931966i \(0.381907\pi\)
\(110\) −2.22931e7 −1.59697
\(111\) 0 0
\(112\) 6.45709e6 0.434284
\(113\) 1.24028e7 0.808624 0.404312 0.914621i \(-0.367511\pi\)
0.404312 + 0.914621i \(0.367511\pi\)
\(114\) 0 0
\(115\) 5.86271e6 0.359464
\(116\) 4.57539e6 0.272160
\(117\) 0 0
\(118\) 2.09651e6 0.117465
\(119\) 1.70075e7 0.925182
\(120\) 0 0
\(121\) 1.06704e7 0.547558
\(122\) 2.84611e7 1.41903
\(123\) 0 0
\(124\) 5.30365e6 0.249804
\(125\) 754706. 0.0345615
\(126\) 0 0
\(127\) 9.12926e6 0.395478 0.197739 0.980255i \(-0.436640\pi\)
0.197739 + 0.980255i \(0.436640\pi\)
\(128\) −1.50705e7 −0.635172
\(129\) 0 0
\(130\) −2.13630e7 −0.852828
\(131\) −1.91545e7 −0.744426 −0.372213 0.928147i \(-0.621401\pi\)
−0.372213 + 0.928147i \(0.621401\pi\)
\(132\) 0 0
\(133\) −2.11393e7 −0.779131
\(134\) 4.71028e7 1.69114
\(135\) 0 0
\(136\) −5.21264e7 −1.77693
\(137\) 1.11699e7 0.371132 0.185566 0.982632i \(-0.440588\pi\)
0.185566 + 0.982632i \(0.440588\pi\)
\(138\) 0 0
\(139\) 4.22972e7 1.33586 0.667928 0.744226i \(-0.267182\pi\)
0.667928 + 0.744226i \(0.267182\pi\)
\(140\) 4.78449e6 0.147363
\(141\) 0 0
\(142\) 1.53035e7 0.448518
\(143\) 2.88994e7 0.826442
\(144\) 0 0
\(145\) −7.64613e7 −2.08283
\(146\) 1.41836e7 0.377183
\(147\) 0 0
\(148\) −7.61027e6 −0.192967
\(149\) −5.92562e7 −1.46751 −0.733757 0.679412i \(-0.762235\pi\)
−0.733757 + 0.679412i \(0.762235\pi\)
\(150\) 0 0
\(151\) 7.21512e7 1.70539 0.852697 0.522406i \(-0.174966\pi\)
0.852697 + 0.522406i \(0.174966\pi\)
\(152\) 6.47899e7 1.49642
\(153\) 0 0
\(154\) 2.83417e7 0.625321
\(155\) −8.86316e7 −1.91174
\(156\) 0 0
\(157\) −5.11820e7 −1.05552 −0.527762 0.849392i \(-0.676969\pi\)
−0.527762 + 0.849392i \(0.676969\pi\)
\(158\) −7.28421e7 −1.46921
\(159\) 0 0
\(160\) −2.70291e7 −0.521690
\(161\) −7.45339e6 −0.140755
\(162\) 0 0
\(163\) 3.02638e7 0.547352 0.273676 0.961822i \(-0.411760\pi\)
0.273676 + 0.961822i \(0.411760\pi\)
\(164\) −4.99736e6 −0.0884682
\(165\) 0 0
\(166\) 2.07557e7 0.352175
\(167\) −5.48855e7 −0.911906 −0.455953 0.890004i \(-0.650702\pi\)
−0.455953 + 0.890004i \(0.650702\pi\)
\(168\) 0 0
\(169\) −3.50548e7 −0.558655
\(170\) 1.36561e8 2.13184
\(171\) 0 0
\(172\) 4.70814e6 0.0705504
\(173\) 9.94511e7 1.46032 0.730161 0.683276i \(-0.239445\pi\)
0.730161 + 0.683276i \(0.239445\pi\)
\(174\) 0 0
\(175\) −4.04577e7 −0.570647
\(176\) −7.01371e7 −0.969736
\(177\) 0 0
\(178\) 5.63637e7 0.749083
\(179\) −1.27756e8 −1.66493 −0.832465 0.554078i \(-0.813071\pi\)
−0.832465 + 0.554078i \(0.813071\pi\)
\(180\) 0 0
\(181\) 5.65804e7 0.709237 0.354618 0.935011i \(-0.384611\pi\)
0.354618 + 0.935011i \(0.384611\pi\)
\(182\) 2.71593e7 0.333940
\(183\) 0 0
\(184\) 2.28439e7 0.270338
\(185\) 1.27179e8 1.47677
\(186\) 0 0
\(187\) −1.84736e8 −2.06589
\(188\) 2.95605e7 0.324459
\(189\) 0 0
\(190\) −1.69737e8 −1.79531
\(191\) 1.10143e8 1.14378 0.571888 0.820331i \(-0.306211\pi\)
0.571888 + 0.820331i \(0.306211\pi\)
\(192\) 0 0
\(193\) 8.48163e7 0.849236 0.424618 0.905373i \(-0.360408\pi\)
0.424618 + 0.905373i \(0.360408\pi\)
\(194\) −1.53804e8 −1.51239
\(195\) 0 0
\(196\) 1.35150e7 0.128209
\(197\) −8.45072e7 −0.787520 −0.393760 0.919213i \(-0.628826\pi\)
−0.393760 + 0.919213i \(0.628826\pi\)
\(198\) 0 0
\(199\) −1.14880e8 −1.03337 −0.516686 0.856175i \(-0.672835\pi\)
−0.516686 + 0.856175i \(0.672835\pi\)
\(200\) 1.23999e8 1.09600
\(201\) 0 0
\(202\) −7.25360e7 −0.619190
\(203\) 9.72070e7 0.815570
\(204\) 0 0
\(205\) 8.35132e7 0.677043
\(206\) −5.53129e7 −0.440850
\(207\) 0 0
\(208\) −6.72110e7 −0.517868
\(209\) 2.29616e8 1.73976
\(210\) 0 0
\(211\) −2.23391e8 −1.63711 −0.818555 0.574428i \(-0.805225\pi\)
−0.818555 + 0.574428i \(0.805225\pi\)
\(212\) −8.24214e6 −0.0594107
\(213\) 0 0
\(214\) 2.02303e8 1.41109
\(215\) −7.86798e7 −0.539919
\(216\) 0 0
\(217\) 1.12679e8 0.748575
\(218\) −1.00075e8 −0.654229
\(219\) 0 0
\(220\) −5.19693e7 −0.329054
\(221\) −1.77029e8 −1.10325
\(222\) 0 0
\(223\) −1.25281e8 −0.756519 −0.378259 0.925700i \(-0.623477\pi\)
−0.378259 + 0.925700i \(0.623477\pi\)
\(224\) 3.43627e7 0.204277
\(225\) 0 0
\(226\) −1.26608e8 −0.729596
\(227\) 1.69633e7 0.0962540 0.0481270 0.998841i \(-0.484675\pi\)
0.0481270 + 0.998841i \(0.484675\pi\)
\(228\) 0 0
\(229\) −9.47495e7 −0.521378 −0.260689 0.965423i \(-0.583950\pi\)
−0.260689 + 0.965423i \(0.583950\pi\)
\(230\) −5.98465e7 −0.324333
\(231\) 0 0
\(232\) −2.97929e8 −1.56641
\(233\) 1.23601e8 0.640143 0.320072 0.947393i \(-0.396293\pi\)
0.320072 + 0.947393i \(0.396293\pi\)
\(234\) 0 0
\(235\) −4.93999e8 −2.48307
\(236\) 4.88735e6 0.0242037
\(237\) 0 0
\(238\) −1.73613e8 −0.834762
\(239\) −1.84669e8 −0.874985 −0.437493 0.899222i \(-0.644133\pi\)
−0.437493 + 0.899222i \(0.644133\pi\)
\(240\) 0 0
\(241\) −3.10641e8 −1.42955 −0.714775 0.699354i \(-0.753471\pi\)
−0.714775 + 0.699354i \(0.753471\pi\)
\(242\) −1.08923e8 −0.494044
\(243\) 0 0
\(244\) 6.63480e7 0.292391
\(245\) −2.25855e8 −0.981181
\(246\) 0 0
\(247\) 2.20037e8 0.929086
\(248\) −3.45351e8 −1.43774
\(249\) 0 0
\(250\) −7.70404e6 −0.0311838
\(251\) −1.38239e8 −0.551788 −0.275894 0.961188i \(-0.588974\pi\)
−0.275894 + 0.961188i \(0.588974\pi\)
\(252\) 0 0
\(253\) 8.09589e7 0.314299
\(254\) −9.31915e7 −0.356827
\(255\) 0 0
\(256\) −1.44221e8 −0.537264
\(257\) 1.25743e8 0.462080 0.231040 0.972944i \(-0.425787\pi\)
0.231040 + 0.972944i \(0.425787\pi\)
\(258\) 0 0
\(259\) −1.61685e8 −0.578257
\(260\) −4.98012e7 −0.175725
\(261\) 0 0
\(262\) 1.95529e8 0.671672
\(263\) 2.91293e8 0.987380 0.493690 0.869638i \(-0.335648\pi\)
0.493690 + 0.869638i \(0.335648\pi\)
\(264\) 0 0
\(265\) 1.37738e8 0.454667
\(266\) 2.15790e8 0.702985
\(267\) 0 0
\(268\) 1.09805e8 0.348459
\(269\) 7.18507e7 0.225060 0.112530 0.993648i \(-0.464105\pi\)
0.112530 + 0.993648i \(0.464105\pi\)
\(270\) 0 0
\(271\) 2.99889e7 0.0915310 0.0457655 0.998952i \(-0.485427\pi\)
0.0457655 + 0.998952i \(0.485427\pi\)
\(272\) 4.29639e8 1.29453
\(273\) 0 0
\(274\) −1.14023e8 −0.334861
\(275\) 4.39452e8 1.27423
\(276\) 0 0
\(277\) 3.52435e8 0.996322 0.498161 0.867085i \(-0.334009\pi\)
0.498161 + 0.867085i \(0.334009\pi\)
\(278\) −4.31769e8 −1.20530
\(279\) 0 0
\(280\) −3.11546e8 −0.848142
\(281\) −3.70901e8 −0.997209 −0.498605 0.866830i \(-0.666154\pi\)
−0.498605 + 0.866830i \(0.666154\pi\)
\(282\) 0 0
\(283\) 1.18723e8 0.311373 0.155687 0.987807i \(-0.450241\pi\)
0.155687 + 0.987807i \(0.450241\pi\)
\(284\) 3.56752e7 0.0924170
\(285\) 0 0
\(286\) −2.95005e8 −0.745673
\(287\) −1.06172e8 −0.265109
\(288\) 0 0
\(289\) 7.21302e8 1.75782
\(290\) 7.80517e8 1.87927
\(291\) 0 0
\(292\) 3.30647e7 0.0777184
\(293\) −6.18045e8 −1.43544 −0.717718 0.696334i \(-0.754813\pi\)
−0.717718 + 0.696334i \(0.754813\pi\)
\(294\) 0 0
\(295\) −8.16747e7 −0.185230
\(296\) 4.95548e8 1.11062
\(297\) 0 0
\(298\) 6.04888e8 1.32409
\(299\) 7.75814e7 0.167845
\(300\) 0 0
\(301\) 1.00027e8 0.211415
\(302\) −7.36520e8 −1.53872
\(303\) 0 0
\(304\) −5.34016e8 −1.09018
\(305\) −1.10877e9 −2.23765
\(306\) 0 0
\(307\) −7.10474e8 −1.40141 −0.700703 0.713453i \(-0.747130\pi\)
−0.700703 + 0.713453i \(0.747130\pi\)
\(308\) 6.60697e7 0.128847
\(309\) 0 0
\(310\) 9.04752e8 1.72490
\(311\) −8.99410e8 −1.69549 −0.847747 0.530401i \(-0.822042\pi\)
−0.847747 + 0.530401i \(0.822042\pi\)
\(312\) 0 0
\(313\) −1.68969e8 −0.311459 −0.155729 0.987800i \(-0.549773\pi\)
−0.155729 + 0.987800i \(0.549773\pi\)
\(314\) 5.22466e8 0.952366
\(315\) 0 0
\(316\) −1.69808e8 −0.302729
\(317\) −6.45496e8 −1.13812 −0.569058 0.822297i \(-0.692692\pi\)
−0.569058 + 0.822297i \(0.692692\pi\)
\(318\) 0 0
\(319\) −1.05586e9 −1.82113
\(320\) 9.26030e8 1.57979
\(321\) 0 0
\(322\) 7.60842e7 0.126999
\(323\) −1.40656e9 −2.32247
\(324\) 0 0
\(325\) 4.21119e8 0.680476
\(326\) −3.08933e8 −0.493858
\(327\) 0 0
\(328\) 3.25407e8 0.509176
\(329\) 6.28032e8 0.972291
\(330\) 0 0
\(331\) −1.03251e9 −1.56493 −0.782467 0.622692i \(-0.786039\pi\)
−0.782467 + 0.622692i \(0.786039\pi\)
\(332\) 4.83853e7 0.0725655
\(333\) 0 0
\(334\) 5.60272e8 0.822784
\(335\) −1.83501e9 −2.66674
\(336\) 0 0
\(337\) −2.83276e8 −0.403186 −0.201593 0.979469i \(-0.564612\pi\)
−0.201593 + 0.979469i \(0.564612\pi\)
\(338\) 3.57839e8 0.504057
\(339\) 0 0
\(340\) 3.18349e8 0.439265
\(341\) −1.22393e9 −1.67153
\(342\) 0 0
\(343\) 7.03499e8 0.941313
\(344\) −3.06574e8 −0.406051
\(345\) 0 0
\(346\) −1.01520e9 −1.31760
\(347\) −3.08826e8 −0.396790 −0.198395 0.980122i \(-0.563573\pi\)
−0.198395 + 0.980122i \(0.563573\pi\)
\(348\) 0 0
\(349\) −9.76048e8 −1.22909 −0.614543 0.788884i \(-0.710659\pi\)
−0.614543 + 0.788884i \(0.710659\pi\)
\(350\) 4.12992e8 0.514877
\(351\) 0 0
\(352\) −3.73249e8 −0.456141
\(353\) 6.80666e8 0.823612 0.411806 0.911271i \(-0.364898\pi\)
0.411806 + 0.911271i \(0.364898\pi\)
\(354\) 0 0
\(355\) −5.96184e8 −0.707263
\(356\) 1.31394e8 0.154348
\(357\) 0 0
\(358\) 1.30413e9 1.50221
\(359\) −8.85168e8 −1.00971 −0.504853 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(360\) 0 0
\(361\) 8.54398e8 0.955840
\(362\) −5.77573e8 −0.639922
\(363\) 0 0
\(364\) 6.33134e7 0.0688083
\(365\) −5.52558e8 −0.594776
\(366\) 0 0
\(367\) 1.33637e9 1.41122 0.705610 0.708601i \(-0.250673\pi\)
0.705610 + 0.708601i \(0.250673\pi\)
\(368\) −1.88285e8 −0.196947
\(369\) 0 0
\(370\) −1.29824e9 −1.33244
\(371\) −1.75110e8 −0.178033
\(372\) 0 0
\(373\) 6.67780e8 0.666274 0.333137 0.942879i \(-0.391893\pi\)
0.333137 + 0.942879i \(0.391893\pi\)
\(374\) 1.88579e9 1.86398
\(375\) 0 0
\(376\) −1.92485e9 −1.86741
\(377\) −1.01182e9 −0.972538
\(378\) 0 0
\(379\) 1.22986e9 1.16043 0.580213 0.814465i \(-0.302969\pi\)
0.580213 + 0.814465i \(0.302969\pi\)
\(380\) −3.95688e8 −0.369922
\(381\) 0 0
\(382\) −1.12434e9 −1.03199
\(383\) −1.72139e9 −1.56561 −0.782805 0.622267i \(-0.786212\pi\)
−0.782805 + 0.622267i \(0.786212\pi\)
\(384\) 0 0
\(385\) −1.10412e9 −0.986062
\(386\) −8.65804e8 −0.766239
\(387\) 0 0
\(388\) −3.58547e8 −0.311627
\(389\) 8.54144e8 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(390\) 0 0
\(391\) −4.95931e8 −0.419568
\(392\) −8.80039e8 −0.737905
\(393\) 0 0
\(394\) 8.62649e8 0.710555
\(395\) 2.83775e9 2.31677
\(396\) 0 0
\(397\) 4.64867e8 0.372874 0.186437 0.982467i \(-0.440306\pi\)
0.186437 + 0.982467i \(0.440306\pi\)
\(398\) 1.17269e9 0.932379
\(399\) 0 0
\(400\) −1.02203e9 −0.798461
\(401\) −1.35635e9 −1.05043 −0.525216 0.850969i \(-0.676015\pi\)
−0.525216 + 0.850969i \(0.676015\pi\)
\(402\) 0 0
\(403\) −1.17287e9 −0.892648
\(404\) −1.69095e8 −0.127584
\(405\) 0 0
\(406\) −9.92289e8 −0.735863
\(407\) 1.75623e9 1.29122
\(408\) 0 0
\(409\) −1.50649e9 −1.08876 −0.544382 0.838838i \(-0.683236\pi\)
−0.544382 + 0.838838i \(0.683236\pi\)
\(410\) −8.52502e8 −0.610875
\(411\) 0 0
\(412\) −1.28944e8 −0.0908370
\(413\) 1.03835e8 0.0725301
\(414\) 0 0
\(415\) −8.08589e8 −0.555341
\(416\) −3.57677e8 −0.243593
\(417\) 0 0
\(418\) −2.34392e9 −1.56973
\(419\) 2.73741e9 1.81799 0.908995 0.416808i \(-0.136851\pi\)
0.908995 + 0.416808i \(0.136851\pi\)
\(420\) 0 0
\(421\) −4.08418e8 −0.266758 −0.133379 0.991065i \(-0.542583\pi\)
−0.133379 + 0.991065i \(0.542583\pi\)
\(422\) 2.28038e9 1.47711
\(423\) 0 0
\(424\) 5.36693e8 0.341936
\(425\) −2.69196e9 −1.70101
\(426\) 0 0
\(427\) 1.40961e9 0.876193
\(428\) 4.71606e8 0.290754
\(429\) 0 0
\(430\) 8.03164e8 0.487152
\(431\) 7.95781e8 0.478766 0.239383 0.970925i \(-0.423055\pi\)
0.239383 + 0.970925i \(0.423055\pi\)
\(432\) 0 0
\(433\) −1.70983e9 −1.01215 −0.506076 0.862489i \(-0.668905\pi\)
−0.506076 + 0.862489i \(0.668905\pi\)
\(434\) −1.15023e9 −0.675415
\(435\) 0 0
\(436\) −2.33294e8 −0.134804
\(437\) 6.16412e8 0.353334
\(438\) 0 0
\(439\) 3.43402e9 1.93721 0.968606 0.248602i \(-0.0799712\pi\)
0.968606 + 0.248602i \(0.0799712\pi\)
\(440\) 3.38402e9 1.89386
\(441\) 0 0
\(442\) 1.80711e9 0.995423
\(443\) 8.23020e7 0.0449777 0.0224888 0.999747i \(-0.492841\pi\)
0.0224888 + 0.999747i \(0.492841\pi\)
\(444\) 0 0
\(445\) −2.19579e9 −1.18122
\(446\) 1.27887e9 0.682583
\(447\) 0 0
\(448\) −1.17728e9 −0.618597
\(449\) 1.42868e9 0.744856 0.372428 0.928061i \(-0.378525\pi\)
0.372428 + 0.928061i \(0.378525\pi\)
\(450\) 0 0
\(451\) 1.15324e9 0.591975
\(452\) −2.95147e8 −0.150333
\(453\) 0 0
\(454\) −1.73161e8 −0.0868469
\(455\) −1.05806e9 −0.526586
\(456\) 0 0
\(457\) 1.78047e9 0.872625 0.436312 0.899795i \(-0.356284\pi\)
0.436312 + 0.899795i \(0.356284\pi\)
\(458\) 9.67203e8 0.470423
\(459\) 0 0
\(460\) −1.39513e8 −0.0668287
\(461\) −2.67034e9 −1.26944 −0.634722 0.772741i \(-0.718885\pi\)
−0.634722 + 0.772741i \(0.718885\pi\)
\(462\) 0 0
\(463\) 9.93730e7 0.0465302 0.0232651 0.999729i \(-0.492594\pi\)
0.0232651 + 0.999729i \(0.492594\pi\)
\(464\) 2.45561e9 1.14116
\(465\) 0 0
\(466\) −1.26172e9 −0.577581
\(467\) 2.09963e9 0.953967 0.476984 0.878912i \(-0.341730\pi\)
0.476984 + 0.878912i \(0.341730\pi\)
\(468\) 0 0
\(469\) 2.33289e9 1.04421
\(470\) 5.04274e9 2.24039
\(471\) 0 0
\(472\) −3.18243e8 −0.139304
\(473\) −1.08650e9 −0.472080
\(474\) 0 0
\(475\) 3.34594e9 1.43249
\(476\) −4.04724e8 −0.172002
\(477\) 0 0
\(478\) 1.88510e9 0.789471
\(479\) −3.04269e8 −0.126498 −0.0632490 0.997998i \(-0.520146\pi\)
−0.0632490 + 0.997998i \(0.520146\pi\)
\(480\) 0 0
\(481\) 1.68296e9 0.689550
\(482\) 3.17103e9 1.28984
\(483\) 0 0
\(484\) −2.53920e8 −0.101798
\(485\) 5.99184e9 2.38487
\(486\) 0 0
\(487\) −8.62328e8 −0.338315 −0.169157 0.985589i \(-0.554105\pi\)
−0.169157 + 0.985589i \(0.554105\pi\)
\(488\) −4.32030e9 −1.68284
\(489\) 0 0
\(490\) 2.30553e9 0.885288
\(491\) 1.09588e9 0.417811 0.208905 0.977936i \(-0.433010\pi\)
0.208905 + 0.977936i \(0.433010\pi\)
\(492\) 0 0
\(493\) 6.46792e9 2.43109
\(494\) −2.24613e9 −0.838284
\(495\) 0 0
\(496\) 2.84647e9 1.04742
\(497\) 7.57942e8 0.276942
\(498\) 0 0
\(499\) 2.67449e9 0.963582 0.481791 0.876286i \(-0.339986\pi\)
0.481791 + 0.876286i \(0.339986\pi\)
\(500\) −1.79595e7 −0.00642540
\(501\) 0 0
\(502\) 1.41114e9 0.497861
\(503\) 4.00214e9 1.40218 0.701091 0.713072i \(-0.252697\pi\)
0.701091 + 0.713072i \(0.252697\pi\)
\(504\) 0 0
\(505\) 2.82582e9 0.976394
\(506\) −8.26428e8 −0.283582
\(507\) 0 0
\(508\) −2.17247e8 −0.0735241
\(509\) −1.64260e9 −0.552102 −0.276051 0.961143i \(-0.589026\pi\)
−0.276051 + 0.961143i \(0.589026\pi\)
\(510\) 0 0
\(511\) 7.02480e8 0.232895
\(512\) 3.40122e9 1.11993
\(513\) 0 0
\(514\) −1.28358e9 −0.416920
\(515\) 2.15485e9 0.695171
\(516\) 0 0
\(517\) −6.82170e9 −2.17108
\(518\) 1.65048e9 0.521743
\(519\) 0 0
\(520\) 3.24284e9 1.01138
\(521\) −7.57722e8 −0.234735 −0.117368 0.993089i \(-0.537446\pi\)
−0.117368 + 0.993089i \(0.537446\pi\)
\(522\) 0 0
\(523\) 7.00512e8 0.214121 0.107061 0.994252i \(-0.465856\pi\)
0.107061 + 0.994252i \(0.465856\pi\)
\(524\) 4.55815e8 0.138398
\(525\) 0 0
\(526\) −2.97352e9 −0.890882
\(527\) 7.49741e9 2.23138
\(528\) 0 0
\(529\) −3.18749e9 −0.936168
\(530\) −1.40603e9 −0.410232
\(531\) 0 0
\(532\) 5.03047e8 0.144850
\(533\) 1.10513e9 0.316132
\(534\) 0 0
\(535\) −7.88122e9 −2.22513
\(536\) −7.15006e9 −2.00555
\(537\) 0 0
\(538\) −7.33452e8 −0.203064
\(539\) −3.11887e9 −0.857899
\(540\) 0 0
\(541\) −1.39710e9 −0.379346 −0.189673 0.981847i \(-0.560743\pi\)
−0.189673 + 0.981847i \(0.560743\pi\)
\(542\) −3.06127e8 −0.0825855
\(543\) 0 0
\(544\) 2.28641e9 0.608918
\(545\) 3.89869e9 1.03165
\(546\) 0 0
\(547\) −1.96444e8 −0.0513197 −0.0256598 0.999671i \(-0.508169\pi\)
−0.0256598 + 0.999671i \(0.508169\pi\)
\(548\) −2.65808e8 −0.0689979
\(549\) 0 0
\(550\) −4.48593e9 −1.14970
\(551\) −8.03923e9 −2.04731
\(552\) 0 0
\(553\) −3.60769e9 −0.907175
\(554\) −3.59766e9 −0.898950
\(555\) 0 0
\(556\) −1.00653e9 −0.248352
\(557\) −2.95103e9 −0.723571 −0.361785 0.932261i \(-0.617833\pi\)
−0.361785 + 0.932261i \(0.617833\pi\)
\(558\) 0 0
\(559\) −1.04117e9 −0.252105
\(560\) 2.56784e9 0.617889
\(561\) 0 0
\(562\) 3.78616e9 0.899750
\(563\) 9.45756e7 0.0223357 0.0111679 0.999938i \(-0.496445\pi\)
0.0111679 + 0.999938i \(0.496445\pi\)
\(564\) 0 0
\(565\) 4.93234e9 1.15049
\(566\) −1.21192e9 −0.280942
\(567\) 0 0
\(568\) −2.32301e9 −0.531903
\(569\) −6.34844e8 −0.144469 −0.0722344 0.997388i \(-0.523013\pi\)
−0.0722344 + 0.997388i \(0.523013\pi\)
\(570\) 0 0
\(571\) 6.22435e9 1.39916 0.699580 0.714554i \(-0.253370\pi\)
0.699580 + 0.714554i \(0.253370\pi\)
\(572\) −6.87711e8 −0.153646
\(573\) 0 0
\(574\) 1.08381e9 0.239199
\(575\) 1.17972e9 0.258787
\(576\) 0 0
\(577\) 5.59231e9 1.21193 0.605963 0.795493i \(-0.292788\pi\)
0.605963 + 0.795493i \(0.292788\pi\)
\(578\) −7.36305e9 −1.58603
\(579\) 0 0
\(580\) 1.81953e9 0.387223
\(581\) 1.02798e9 0.217454
\(582\) 0 0
\(583\) 1.90204e9 0.397540
\(584\) −2.15303e9 −0.447306
\(585\) 0 0
\(586\) 6.30901e9 1.29515
\(587\) −5.92527e9 −1.20913 −0.604567 0.796554i \(-0.706654\pi\)
−0.604567 + 0.796554i \(0.706654\pi\)
\(588\) 0 0
\(589\) −9.31883e9 −1.87913
\(590\) 8.33735e8 0.167127
\(591\) 0 0
\(592\) −4.08444e9 −0.809108
\(593\) 4.66790e9 0.919242 0.459621 0.888115i \(-0.347985\pi\)
0.459621 + 0.888115i \(0.347985\pi\)
\(594\) 0 0
\(595\) 6.76352e9 1.31633
\(596\) 1.41011e9 0.272828
\(597\) 0 0
\(598\) −7.91951e8 −0.151441
\(599\) −2.89602e9 −0.550563 −0.275282 0.961364i \(-0.588771\pi\)
−0.275282 + 0.961364i \(0.588771\pi\)
\(600\) 0 0
\(601\) 4.95609e9 0.931276 0.465638 0.884975i \(-0.345825\pi\)
0.465638 + 0.884975i \(0.345825\pi\)
\(602\) −1.02108e9 −0.190753
\(603\) 0 0
\(604\) −1.71696e9 −0.317053
\(605\) 4.24337e9 0.779053
\(606\) 0 0
\(607\) −2.16921e9 −0.393678 −0.196839 0.980436i \(-0.563068\pi\)
−0.196839 + 0.980436i \(0.563068\pi\)
\(608\) −2.84187e9 −0.512793
\(609\) 0 0
\(610\) 1.13183e10 2.01896
\(611\) −6.53711e9 −1.15942
\(612\) 0 0
\(613\) −2.26724e9 −0.397544 −0.198772 0.980046i \(-0.563695\pi\)
−0.198772 + 0.980046i \(0.563695\pi\)
\(614\) 7.25252e9 1.26444
\(615\) 0 0
\(616\) −4.30218e9 −0.741576
\(617\) −3.64399e9 −0.624567 −0.312284 0.949989i \(-0.601094\pi\)
−0.312284 + 0.949989i \(0.601094\pi\)
\(618\) 0 0
\(619\) 8.80200e9 1.49164 0.745820 0.666147i \(-0.232058\pi\)
0.745820 + 0.666147i \(0.232058\pi\)
\(620\) 2.10914e9 0.355415
\(621\) 0 0
\(622\) 9.18118e9 1.52979
\(623\) 2.79155e9 0.462528
\(624\) 0 0
\(625\) −5.95165e9 −0.975118
\(626\) 1.72483e9 0.281019
\(627\) 0 0
\(628\) 1.21796e9 0.196235
\(629\) −1.07581e10 −1.72369
\(630\) 0 0
\(631\) 3.58813e9 0.568546 0.284273 0.958743i \(-0.408248\pi\)
0.284273 + 0.958743i \(0.408248\pi\)
\(632\) 1.10572e10 1.74235
\(633\) 0 0
\(634\) 6.58923e9 1.02689
\(635\) 3.63051e9 0.562677
\(636\) 0 0
\(637\) −2.98875e9 −0.458143
\(638\) 1.07783e10 1.64315
\(639\) 0 0
\(640\) −5.99319e9 −0.903708
\(641\) −5.09433e9 −0.763984 −0.381992 0.924166i \(-0.624762\pi\)
−0.381992 + 0.924166i \(0.624762\pi\)
\(642\) 0 0
\(643\) −6.13203e9 −0.909633 −0.454816 0.890585i \(-0.650295\pi\)
−0.454816 + 0.890585i \(0.650295\pi\)
\(644\) 1.77366e8 0.0261680
\(645\) 0 0
\(646\) 1.43582e10 2.09549
\(647\) 7.78100e9 1.12946 0.564729 0.825276i \(-0.308981\pi\)
0.564729 + 0.825276i \(0.308981\pi\)
\(648\) 0 0
\(649\) −1.12786e9 −0.161956
\(650\) −4.29878e9 −0.613972
\(651\) 0 0
\(652\) −7.20179e8 −0.101759
\(653\) 6.82855e8 0.0959693 0.0479846 0.998848i \(-0.484720\pi\)
0.0479846 + 0.998848i \(0.484720\pi\)
\(654\) 0 0
\(655\) −7.61733e9 −1.05915
\(656\) −2.68209e9 −0.370945
\(657\) 0 0
\(658\) −6.41095e9 −0.877267
\(659\) −6.06750e8 −0.0825868 −0.0412934 0.999147i \(-0.513148\pi\)
−0.0412934 + 0.999147i \(0.513148\pi\)
\(660\) 0 0
\(661\) −1.22400e9 −0.164846 −0.0824228 0.996597i \(-0.526266\pi\)
−0.0824228 + 0.996597i \(0.526266\pi\)
\(662\) 1.05399e10 1.41199
\(663\) 0 0
\(664\) −3.15064e9 −0.417649
\(665\) −8.40665e9 −1.10853
\(666\) 0 0
\(667\) −2.83450e9 −0.369859
\(668\) 1.30610e9 0.169534
\(669\) 0 0
\(670\) 1.87318e10 2.40612
\(671\) −1.53112e10 −1.95650
\(672\) 0 0
\(673\) −8.76801e9 −1.10879 −0.554394 0.832254i \(-0.687050\pi\)
−0.554394 + 0.832254i \(0.687050\pi\)
\(674\) 2.89168e9 0.363782
\(675\) 0 0
\(676\) 8.34189e8 0.103861
\(677\) −3.66670e9 −0.454166 −0.227083 0.973875i \(-0.572919\pi\)
−0.227083 + 0.973875i \(0.572919\pi\)
\(678\) 0 0
\(679\) −7.61755e9 −0.933837
\(680\) −2.07295e10 −2.52818
\(681\) 0 0
\(682\) 1.24938e10 1.50817
\(683\) 2.41403e8 0.0289915 0.0144957 0.999895i \(-0.495386\pi\)
0.0144957 + 0.999895i \(0.495386\pi\)
\(684\) 0 0
\(685\) 4.44203e9 0.528038
\(686\) −7.18132e9 −0.849317
\(687\) 0 0
\(688\) 2.52686e9 0.295816
\(689\) 1.82269e9 0.212298
\(690\) 0 0
\(691\) 5.87544e9 0.677434 0.338717 0.940888i \(-0.390007\pi\)
0.338717 + 0.940888i \(0.390007\pi\)
\(692\) −2.36661e9 −0.271491
\(693\) 0 0
\(694\) 3.15249e9 0.358011
\(695\) 1.68206e10 1.90062
\(696\) 0 0
\(697\) −7.06444e9 −0.790247
\(698\) 9.96349e9 1.10896
\(699\) 0 0
\(700\) 9.62760e8 0.106090
\(701\) 1.02557e10 1.12449 0.562243 0.826972i \(-0.309939\pi\)
0.562243 + 0.826972i \(0.309939\pi\)
\(702\) 0 0
\(703\) 1.33717e10 1.45159
\(704\) 1.27877e10 1.38130
\(705\) 0 0
\(706\) −6.94824e9 −0.743119
\(707\) −3.59253e9 −0.382325
\(708\) 0 0
\(709\) −5.18318e9 −0.546178 −0.273089 0.961989i \(-0.588045\pi\)
−0.273089 + 0.961989i \(0.588045\pi\)
\(710\) 6.08585e9 0.638141
\(711\) 0 0
\(712\) −8.55583e9 −0.888346
\(713\) −3.28567e9 −0.339477
\(714\) 0 0
\(715\) 1.14927e10 1.17584
\(716\) 3.04018e9 0.309530
\(717\) 0 0
\(718\) 9.03579e9 0.911026
\(719\) −1.29607e10 −1.30040 −0.650202 0.759762i \(-0.725316\pi\)
−0.650202 + 0.759762i \(0.725316\pi\)
\(720\) 0 0
\(721\) −2.73951e9 −0.272207
\(722\) −8.72170e9 −0.862424
\(723\) 0 0
\(724\) −1.34643e9 −0.131856
\(725\) −1.53859e10 −1.49948
\(726\) 0 0
\(727\) 4.17261e9 0.402752 0.201376 0.979514i \(-0.435459\pi\)
0.201376 + 0.979514i \(0.435459\pi\)
\(728\) −4.12269e9 −0.396024
\(729\) 0 0
\(730\) 5.64052e9 0.536647
\(731\) 6.65558e9 0.630195
\(732\) 0 0
\(733\) −1.52037e10 −1.42589 −0.712944 0.701221i \(-0.752638\pi\)
−0.712944 + 0.701221i \(0.752638\pi\)
\(734\) −1.36416e10 −1.27330
\(735\) 0 0
\(736\) −1.00200e9 −0.0926392
\(737\) −2.53399e10 −2.33168
\(738\) 0 0
\(739\) −7.33278e9 −0.668364 −0.334182 0.942509i \(-0.608460\pi\)
−0.334182 + 0.942509i \(0.608460\pi\)
\(740\) −3.02644e9 −0.274549
\(741\) 0 0
\(742\) 1.78752e9 0.160634
\(743\) −6.55520e9 −0.586307 −0.293154 0.956065i \(-0.594705\pi\)
−0.293154 + 0.956065i \(0.594705\pi\)
\(744\) 0 0
\(745\) −2.35649e10 −2.08794
\(746\) −6.81670e9 −0.601158
\(747\) 0 0
\(748\) 4.39612e9 0.384073
\(749\) 1.00196e10 0.871290
\(750\) 0 0
\(751\) −7.42395e9 −0.639581 −0.319790 0.947488i \(-0.603612\pi\)
−0.319790 + 0.947488i \(0.603612\pi\)
\(752\) 1.58652e10 1.36045
\(753\) 0 0
\(754\) 1.03286e10 0.877490
\(755\) 2.86930e10 2.42639
\(756\) 0 0
\(757\) −1.49414e10 −1.25186 −0.625928 0.779880i \(-0.715280\pi\)
−0.625928 + 0.779880i \(0.715280\pi\)
\(758\) −1.25544e10 −1.04701
\(759\) 0 0
\(760\) 2.57655e10 2.12908
\(761\) 1.92886e10 1.58655 0.793276 0.608862i \(-0.208374\pi\)
0.793276 + 0.608862i \(0.208374\pi\)
\(762\) 0 0
\(763\) −4.95649e9 −0.403960
\(764\) −2.62105e9 −0.212642
\(765\) 0 0
\(766\) 1.75720e10 1.41260
\(767\) −1.08080e9 −0.0864894
\(768\) 0 0
\(769\) 1.59930e10 1.26820 0.634099 0.773252i \(-0.281371\pi\)
0.634099 + 0.773252i \(0.281371\pi\)
\(770\) 1.12709e10 0.889692
\(771\) 0 0
\(772\) −2.01835e9 −0.157883
\(773\) −2.25204e10 −1.75367 −0.876833 0.480794i \(-0.840348\pi\)
−0.876833 + 0.480794i \(0.840348\pi\)
\(774\) 0 0
\(775\) −1.78349e10 −1.37631
\(776\) 2.33470e10 1.79356
\(777\) 0 0
\(778\) −8.71910e9 −0.663809
\(779\) 8.78067e9 0.665498
\(780\) 0 0
\(781\) −8.23278e9 −0.618398
\(782\) 5.06246e9 0.378563
\(783\) 0 0
\(784\) 7.25352e9 0.537579
\(785\) −2.03539e10 −1.50178
\(786\) 0 0
\(787\) −1.72331e10 −1.26024 −0.630119 0.776498i \(-0.716994\pi\)
−0.630119 + 0.776498i \(0.716994\pi\)
\(788\) 2.01099e9 0.146409
\(789\) 0 0
\(790\) −2.89677e10 −2.09035
\(791\) −6.27059e9 −0.450496
\(792\) 0 0
\(793\) −1.46724e10 −1.04483
\(794\) −4.74537e9 −0.336432
\(795\) 0 0
\(796\) 2.73376e9 0.192116
\(797\) 1.26332e10 0.883911 0.441955 0.897037i \(-0.354285\pi\)
0.441955 + 0.897037i \(0.354285\pi\)
\(798\) 0 0
\(799\) 4.17878e10 2.89825
\(800\) −5.43894e9 −0.375577
\(801\) 0 0
\(802\) 1.38457e10 0.947771
\(803\) −7.63035e9 −0.520044
\(804\) 0 0
\(805\) −2.96405e9 −0.200263
\(806\) 1.19726e10 0.805408
\(807\) 0 0
\(808\) 1.10107e10 0.734305
\(809\) −9.80378e9 −0.650989 −0.325495 0.945544i \(-0.605531\pi\)
−0.325495 + 0.945544i \(0.605531\pi\)
\(810\) 0 0
\(811\) −1.10475e10 −0.727262 −0.363631 0.931543i \(-0.618463\pi\)
−0.363631 + 0.931543i \(0.618463\pi\)
\(812\) −2.31321e9 −0.151624
\(813\) 0 0
\(814\) −1.79276e10 −1.16503
\(815\) 1.20352e10 0.778759
\(816\) 0 0
\(817\) −8.27249e9 −0.530712
\(818\) 1.53782e10 0.982357
\(819\) 0 0
\(820\) −1.98734e9 −0.125870
\(821\) 1.50386e10 0.948433 0.474217 0.880408i \(-0.342731\pi\)
0.474217 + 0.880408i \(0.342731\pi\)
\(822\) 0 0
\(823\) −1.10764e10 −0.692625 −0.346313 0.938119i \(-0.612566\pi\)
−0.346313 + 0.938119i \(0.612566\pi\)
\(824\) 8.39631e9 0.522809
\(825\) 0 0
\(826\) −1.05995e9 −0.0654416
\(827\) −3.12849e10 −1.92338 −0.961691 0.274137i \(-0.911608\pi\)
−0.961691 + 0.274137i \(0.911608\pi\)
\(828\) 0 0
\(829\) 1.84399e10 1.12413 0.562067 0.827092i \(-0.310006\pi\)
0.562067 + 0.827092i \(0.310006\pi\)
\(830\) 8.25408e9 0.501066
\(831\) 0 0
\(832\) 1.22542e10 0.737654
\(833\) 1.91053e10 1.14524
\(834\) 0 0
\(835\) −2.18268e10 −1.29744
\(836\) −5.46411e9 −0.323443
\(837\) 0 0
\(838\) −2.79435e10 −1.64031
\(839\) −2.82373e10 −1.65065 −0.825327 0.564654i \(-0.809009\pi\)
−0.825327 + 0.564654i \(0.809009\pi\)
\(840\) 0 0
\(841\) 1.97177e10 1.14306
\(842\) 4.16913e9 0.240687
\(843\) 0 0
\(844\) 5.31599e9 0.304358
\(845\) −1.39405e10 −0.794841
\(846\) 0 0
\(847\) −5.39469e9 −0.305052
\(848\) −4.42357e9 −0.249108
\(849\) 0 0
\(850\) 2.74795e10 1.53477
\(851\) 4.71465e9 0.262238
\(852\) 0 0
\(853\) −1.66862e10 −0.920523 −0.460261 0.887783i \(-0.652244\pi\)
−0.460261 + 0.887783i \(0.652244\pi\)
\(854\) −1.43893e10 −0.790561
\(855\) 0 0
\(856\) −3.07089e10 −1.67343
\(857\) 2.12716e10 1.15443 0.577215 0.816592i \(-0.304139\pi\)
0.577215 + 0.816592i \(0.304139\pi\)
\(858\) 0 0
\(859\) 2.18013e10 1.17356 0.586781 0.809745i \(-0.300395\pi\)
0.586781 + 0.809745i \(0.300395\pi\)
\(860\) 1.87232e9 0.100377
\(861\) 0 0
\(862\) −8.12333e9 −0.431975
\(863\) 2.57691e10 1.36478 0.682388 0.730990i \(-0.260942\pi\)
0.682388 + 0.730990i \(0.260942\pi\)
\(864\) 0 0
\(865\) 3.95495e10 2.07771
\(866\) 1.74540e10 0.913233
\(867\) 0 0
\(868\) −2.68140e9 −0.139169
\(869\) 3.91868e10 2.02568
\(870\) 0 0
\(871\) −2.42827e10 −1.24518
\(872\) 1.51911e10 0.775858
\(873\) 0 0
\(874\) −6.29233e9 −0.318802
\(875\) −3.81562e8 −0.0192547
\(876\) 0 0
\(877\) −2.96656e10 −1.48510 −0.742549 0.669792i \(-0.766383\pi\)
−0.742549 + 0.669792i \(0.766383\pi\)
\(878\) −3.50545e10 −1.74788
\(879\) 0 0
\(880\) −2.78920e10 −1.37972
\(881\) 2.10977e10 1.03949 0.519745 0.854322i \(-0.326027\pi\)
0.519745 + 0.854322i \(0.326027\pi\)
\(882\) 0 0
\(883\) 2.21889e9 0.108461 0.0542304 0.998528i \(-0.482729\pi\)
0.0542304 + 0.998528i \(0.482729\pi\)
\(884\) 4.21272e9 0.205107
\(885\) 0 0
\(886\) −8.40138e8 −0.0405819
\(887\) 7.16266e9 0.344621 0.172311 0.985043i \(-0.444877\pi\)
0.172311 + 0.985043i \(0.444877\pi\)
\(888\) 0 0
\(889\) −4.61554e9 −0.220326
\(890\) 2.24146e10 1.06578
\(891\) 0 0
\(892\) 2.98129e9 0.140646
\(893\) −5.19397e10 −2.44073
\(894\) 0 0
\(895\) −5.08058e10 −2.36882
\(896\) 7.61927e9 0.353863
\(897\) 0 0
\(898\) −1.45840e10 −0.672060
\(899\) 4.28516e10 1.96702
\(900\) 0 0
\(901\) −1.16514e10 −0.530689
\(902\) −1.17723e10 −0.534120
\(903\) 0 0
\(904\) 1.92187e10 0.865236
\(905\) 2.25008e10 1.00908
\(906\) 0 0
\(907\) −8.53267e9 −0.379717 −0.189858 0.981811i \(-0.560803\pi\)
−0.189858 + 0.981811i \(0.560803\pi\)
\(908\) −4.03670e8 −0.0178948
\(909\) 0 0
\(910\) 1.08007e10 0.475122
\(911\) −2.76004e9 −0.120949 −0.0604744 0.998170i \(-0.519261\pi\)
−0.0604744 + 0.998170i \(0.519261\pi\)
\(912\) 0 0
\(913\) −1.11659e10 −0.485564
\(914\) −1.81750e10 −0.787342
\(915\) 0 0
\(916\) 2.25473e9 0.0969304
\(917\) 9.68408e9 0.414730
\(918\) 0 0
\(919\) 1.13500e10 0.482382 0.241191 0.970478i \(-0.422462\pi\)
0.241191 + 0.970478i \(0.422462\pi\)
\(920\) 9.08451e9 0.384631
\(921\) 0 0
\(922\) 2.72588e10 1.14538
\(923\) −7.88932e9 −0.330243
\(924\) 0 0
\(925\) 2.55915e10 1.06316
\(926\) −1.01440e9 −0.0419827
\(927\) 0 0
\(928\) 1.30681e10 0.536776
\(929\) −4.41532e10 −1.80679 −0.903395 0.428810i \(-0.858933\pi\)
−0.903395 + 0.428810i \(0.858933\pi\)
\(930\) 0 0
\(931\) −2.37467e10 −0.964449
\(932\) −2.94131e9 −0.119010
\(933\) 0 0
\(934\) −2.14330e10 −0.860734
\(935\) −7.34655e10 −2.93929
\(936\) 0 0
\(937\) −3.84649e10 −1.52748 −0.763741 0.645522i \(-0.776640\pi\)
−0.763741 + 0.645522i \(0.776640\pi\)
\(938\) −2.38141e10 −0.942159
\(939\) 0 0
\(940\) 1.17556e10 0.461632
\(941\) −2.95756e10 −1.15710 −0.578549 0.815648i \(-0.696381\pi\)
−0.578549 + 0.815648i \(0.696381\pi\)
\(942\) 0 0
\(943\) 3.09592e9 0.120226
\(944\) 2.62305e9 0.101485
\(945\) 0 0
\(946\) 1.10910e10 0.425943
\(947\) 4.30437e10 1.64697 0.823483 0.567341i \(-0.192028\pi\)
0.823483 + 0.567341i \(0.192028\pi\)
\(948\) 0 0
\(949\) −7.31202e9 −0.277719
\(950\) −3.41553e10 −1.29249
\(951\) 0 0
\(952\) 2.63539e10 0.989954
\(953\) −2.33081e10 −0.872333 −0.436166 0.899866i \(-0.643664\pi\)
−0.436166 + 0.899866i \(0.643664\pi\)
\(954\) 0 0
\(955\) 4.38016e10 1.62734
\(956\) 4.39451e9 0.162670
\(957\) 0 0
\(958\) 3.10598e9 0.114135
\(959\) −5.64726e9 −0.206763
\(960\) 0 0
\(961\) 2.21597e10 0.805438
\(962\) −1.71796e10 −0.622159
\(963\) 0 0
\(964\) 7.39225e9 0.265771
\(965\) 3.37296e10 1.20827
\(966\) 0 0
\(967\) 3.20553e9 0.114001 0.0570003 0.998374i \(-0.481846\pi\)
0.0570003 + 0.998374i \(0.481846\pi\)
\(968\) 1.65342e10 0.585893
\(969\) 0 0
\(970\) −6.11647e10 −2.15179
\(971\) 2.31196e10 0.810424 0.405212 0.914223i \(-0.367198\pi\)
0.405212 + 0.914223i \(0.367198\pi\)
\(972\) 0 0
\(973\) −2.13845e10 −0.744224
\(974\) 8.80265e9 0.305251
\(975\) 0 0
\(976\) 3.56090e10 1.22599
\(977\) −9.17629e9 −0.314801 −0.157400 0.987535i \(-0.550311\pi\)
−0.157400 + 0.987535i \(0.550311\pi\)
\(978\) 0 0
\(979\) −3.03219e10 −1.03280
\(980\) 5.37462e9 0.182413
\(981\) 0 0
\(982\) −1.11868e10 −0.376977
\(983\) 2.25628e10 0.757626 0.378813 0.925473i \(-0.376332\pi\)
0.378813 + 0.925473i \(0.376332\pi\)
\(984\) 0 0
\(985\) −3.36066e10 −1.12046
\(986\) −6.60245e10 −2.19349
\(987\) 0 0
\(988\) −5.23615e9 −0.172728
\(989\) −2.91675e9 −0.0958764
\(990\) 0 0
\(991\) 4.42802e10 1.44528 0.722640 0.691224i \(-0.242928\pi\)
0.722640 + 0.691224i \(0.242928\pi\)
\(992\) 1.51481e10 0.492682
\(993\) 0 0
\(994\) −7.73707e9 −0.249876
\(995\) −4.56851e10 −1.47026
\(996\) 0 0
\(997\) 2.83830e10 0.907036 0.453518 0.891247i \(-0.350169\pi\)
0.453518 + 0.891247i \(0.350169\pi\)
\(998\) −2.73012e10 −0.869409
\(999\) 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.e.1.6 18
3.2 odd 2 177.8.a.d.1.13 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.13 18 3.2 odd 2
531.8.a.e.1.6 18 1.1 even 1 trivial