Properties

Label 177.8.a.d.1.13
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $18$
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: \(0\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(9.20800\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+10.2080 q^{2} +27.0000 q^{3} -23.7967 q^{4} -397.678 q^{5} +275.616 q^{6} -505.577 q^{7} -1549.54 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+10.2080 q^{2} +27.0000 q^{3} -23.7967 q^{4} -397.678 q^{5} +275.616 q^{6} -505.577 q^{7} -1549.54 q^{8} +729.000 q^{9} -4059.50 q^{10} -5491.59 q^{11} -642.512 q^{12} +5262.48 q^{13} -5160.93 q^{14} -10737.3 q^{15} -12771.7 q^{16} +33639.9 q^{17} +7441.63 q^{18} +41812.3 q^{19} +9463.44 q^{20} -13650.6 q^{21} -56058.1 q^{22} -14742.3 q^{23} -41837.6 q^{24} +80022.8 q^{25} +53719.4 q^{26} +19683.0 q^{27} +12031.1 q^{28} +192269. q^{29} -109606. q^{30} -222873. q^{31} +67967.4 q^{32} -148273. q^{33} +343396. q^{34} +201057. q^{35} -17347.8 q^{36} +319803. q^{37} +426820. q^{38} +142087. q^{39} +616218. q^{40} -210002. q^{41} -139345. q^{42} -197848. q^{43} +130682. q^{44} -289907. q^{45} -150490. q^{46} +1.24221e6 q^{47} -344837. q^{48} -567935. q^{49} +816873. q^{50} +908276. q^{51} -125230. q^{52} -346356. q^{53} +200924. q^{54} +2.18388e6 q^{55} +783412. q^{56} +1.12893e6 q^{57} +1.96269e6 q^{58} +205379. q^{59} +255513. q^{60} -2.78811e6 q^{61} -2.27509e6 q^{62} -368566. q^{63} +2.32859e6 q^{64} -2.09277e6 q^{65} -1.51357e6 q^{66} -4.61431e6 q^{67} -800519. q^{68} -398043. q^{69} +2.05239e6 q^{70} +1.49916e6 q^{71} -1.12962e6 q^{72} -1.38946e6 q^{73} +3.26455e6 q^{74} +2.16062e6 q^{75} -994997. q^{76} +2.77642e6 q^{77} +1.45042e6 q^{78} +7.13579e6 q^{79} +5.07904e6 q^{80} +531441. q^{81} -2.14370e6 q^{82} +2.03328e6 q^{83} +324839. q^{84} -1.33778e7 q^{85} -2.01963e6 q^{86} +5.19128e6 q^{87} +8.50944e6 q^{88} +5.52152e6 q^{89} -2.95937e6 q^{90} -2.66059e6 q^{91} +350820. q^{92} -6.01757e6 q^{93} +1.26805e7 q^{94} -1.66278e7 q^{95} +1.83512e6 q^{96} +1.50671e7 q^{97} -5.79748e6 q^{98} -4.00337e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.2080 0.902268 0.451134 0.892456i \(-0.351020\pi\)
0.451134 + 0.892456i \(0.351020\pi\)
\(3\) 27.0000 0.577350
\(4\) −23.7967 −0.185912
\(5\) −397.678 −1.42278 −0.711388 0.702799i \(-0.751933\pi\)
−0.711388 + 0.702799i \(0.751933\pi\)
\(6\) 275.616 0.520925
\(7\) −505.577 −0.557114 −0.278557 0.960420i \(-0.589856\pi\)
−0.278557 + 0.960420i \(0.589856\pi\)
\(8\) −1549.54 −1.07001
\(9\) 729.000 0.333333
\(10\) −4059.50 −1.28373
\(11\) −5491.59 −1.24401 −0.622004 0.783014i \(-0.713681\pi\)
−0.622004 + 0.783014i \(0.713681\pi\)
\(12\) −642.512 −0.107336
\(13\) 5262.48 0.664338 0.332169 0.943220i \(-0.392220\pi\)
0.332169 + 0.943220i \(0.392220\pi\)
\(14\) −5160.93 −0.502666
\(15\) −10737.3 −0.821440
\(16\) −12771.7 −0.779525
\(17\) 33639.9 1.66067 0.830334 0.557265i \(-0.188149\pi\)
0.830334 + 0.557265i \(0.188149\pi\)
\(18\) 7441.63 0.300756
\(19\) 41812.3 1.39851 0.699257 0.714871i \(-0.253514\pi\)
0.699257 + 0.714871i \(0.253514\pi\)
\(20\) 9463.44 0.264511
\(21\) −13650.6 −0.321650
\(22\) −56058.1 −1.12243
\(23\) −14742.3 −0.252650 −0.126325 0.991989i \(-0.540318\pi\)
−0.126325 + 0.991989i \(0.540318\pi\)
\(24\) −41837.6 −0.617771
\(25\) 80022.8 1.02429
\(26\) 53719.4 0.599411
\(27\) 19683.0 0.192450
\(28\) 12031.1 0.103574
\(29\) 192269. 1.46392 0.731960 0.681348i \(-0.238606\pi\)
0.731960 + 0.681348i \(0.238606\pi\)
\(30\) −109606. −0.741159
\(31\) −222873. −1.34367 −0.671833 0.740703i \(-0.734493\pi\)
−0.671833 + 0.740703i \(0.734493\pi\)
\(32\) 67967.4 0.366670
\(33\) −148273. −0.718229
\(34\) 343396. 1.49837
\(35\) 201057. 0.792648
\(36\) −17347.8 −0.0619707
\(37\) 319803. 1.03795 0.518975 0.854789i \(-0.326314\pi\)
0.518975 + 0.854789i \(0.326314\pi\)
\(38\) 426820. 1.26183
\(39\) 142087. 0.383556
\(40\) 616218. 1.52239
\(41\) −210002. −0.475861 −0.237930 0.971282i \(-0.576469\pi\)
−0.237930 + 0.971282i \(0.576469\pi\)
\(42\) −139345. −0.290214
\(43\) −197848. −0.379483 −0.189741 0.981834i \(-0.560765\pi\)
−0.189741 + 0.981834i \(0.560765\pi\)
\(44\) 130682. 0.231276
\(45\) −289907. −0.474259
\(46\) −150490. −0.227958
\(47\) 1.24221e6 1.74523 0.872614 0.488410i \(-0.162423\pi\)
0.872614 + 0.488410i \(0.162423\pi\)
\(48\) −344837. −0.450059
\(49\) −567935. −0.689624
\(50\) 816873. 0.924186
\(51\) 908276. 0.958788
\(52\) −125230. −0.123508
\(53\) −346356. −0.319564 −0.159782 0.987152i \(-0.551079\pi\)
−0.159782 + 0.987152i \(0.551079\pi\)
\(54\) 200924. 0.173642
\(55\) 2.18388e6 1.76995
\(56\) 783412. 0.596118
\(57\) 1.12893e6 0.807432
\(58\) 1.96269e6 1.32085
\(59\) 205379. 0.130189
\(60\) 255513. 0.152716
\(61\) −2.78811e6 −1.57274 −0.786368 0.617758i \(-0.788041\pi\)
−0.786368 + 0.617758i \(0.788041\pi\)
\(62\) −2.27509e6 −1.21235
\(63\) −368566. −0.185705
\(64\) 2.32859e6 1.11036
\(65\) −2.09277e6 −0.945204
\(66\) −1.51357e6 −0.648035
\(67\) −4.61431e6 −1.87432 −0.937162 0.348895i \(-0.886557\pi\)
−0.937162 + 0.348895i \(0.886557\pi\)
\(68\) −800519. −0.308738
\(69\) −398043. −0.145868
\(70\) 2.05239e6 0.715181
\(71\) 1.49916e6 0.497101 0.248550 0.968619i \(-0.420046\pi\)
0.248550 + 0.968619i \(0.420046\pi\)
\(72\) −1.12962e6 −0.356670
\(73\) −1.38946e6 −0.418039 −0.209019 0.977911i \(-0.567027\pi\)
−0.209019 + 0.977911i \(0.567027\pi\)
\(74\) 3.26455e6 0.936510
\(75\) 2.16062e6 0.591375
\(76\) −994997. −0.260000
\(77\) 2.77642e6 0.693055
\(78\) 1.45042e6 0.346070
\(79\) 7.13579e6 1.62835 0.814174 0.580621i \(-0.197190\pi\)
0.814174 + 0.580621i \(0.197190\pi\)
\(80\) 5.07904e6 1.10909
\(81\) 531441. 0.111111
\(82\) −2.14370e6 −0.429354
\(83\) 2.03328e6 0.390322 0.195161 0.980771i \(-0.437477\pi\)
0.195161 + 0.980771i \(0.437477\pi\)
\(84\) 324839. 0.0597986
\(85\) −1.33778e7 −2.36276
\(86\) −2.01963e6 −0.342395
\(87\) 5.19128e6 0.845195
\(88\) 8.50944e6 1.33110
\(89\) 5.52152e6 0.830222 0.415111 0.909771i \(-0.363743\pi\)
0.415111 + 0.909771i \(0.363743\pi\)
\(90\) −2.95937e6 −0.427909
\(91\) −2.66059e6 −0.370112
\(92\) 350820. 0.0469707
\(93\) −6.01757e6 −0.775766
\(94\) 1.26805e7 1.57466
\(95\) −1.66278e7 −1.98977
\(96\) 1.83512e6 0.211697
\(97\) 1.50671e7 1.67621 0.838103 0.545512i \(-0.183665\pi\)
0.838103 + 0.545512i \(0.183665\pi\)
\(98\) −5.79748e6 −0.622226
\(99\) −4.00337e6 −0.414670
\(100\) −1.90428e6 −0.190428
\(101\) −7.10580e6 −0.686260 −0.343130 0.939288i \(-0.611487\pi\)
−0.343130 + 0.939288i \(0.611487\pi\)
\(102\) 9.27168e6 0.865084
\(103\) 5.41858e6 0.488602 0.244301 0.969699i \(-0.421442\pi\)
0.244301 + 0.969699i \(0.421442\pi\)
\(104\) −8.15444e6 −0.710849
\(105\) 5.42853e6 0.457636
\(106\) −3.53560e6 −0.288332
\(107\) 1.98181e7 1.56393 0.781967 0.623320i \(-0.214216\pi\)
0.781967 + 0.623320i \(0.214216\pi\)
\(108\) −468391. −0.0357788
\(109\) 9.80363e6 0.725094 0.362547 0.931966i \(-0.381907\pi\)
0.362547 + 0.931966i \(0.381907\pi\)
\(110\) 2.22931e7 1.59697
\(111\) 8.63468e6 0.599261
\(112\) 6.45709e6 0.434284
\(113\) −1.24028e7 −0.808624 −0.404312 0.914621i \(-0.632489\pi\)
−0.404312 + 0.914621i \(0.632489\pi\)
\(114\) 1.15241e7 0.728520
\(115\) 5.86271e6 0.359464
\(116\) −4.57539e6 −0.272160
\(117\) 3.83635e6 0.221446
\(118\) 2.09651e6 0.117465
\(119\) −1.70075e7 −0.925182
\(120\) 1.66379e7 0.878950
\(121\) 1.06704e7 0.547558
\(122\) −2.84611e7 −1.41903
\(123\) −5.67005e6 −0.274738
\(124\) 5.30365e6 0.249804
\(125\) −754706. −0.0345615
\(126\) −3.76232e6 −0.167555
\(127\) 9.12926e6 0.395478 0.197739 0.980255i \(-0.436640\pi\)
0.197739 + 0.980255i \(0.436640\pi\)
\(128\) 1.50705e7 0.635172
\(129\) −5.34190e6 −0.219095
\(130\) −2.13630e7 −0.852828
\(131\) 1.91545e7 0.744426 0.372213 0.928147i \(-0.378599\pi\)
0.372213 + 0.928147i \(0.378599\pi\)
\(132\) 3.52841e6 0.133527
\(133\) −2.11393e7 −0.779131
\(134\) −4.71028e7 −1.69114
\(135\) −7.82750e6 −0.273813
\(136\) −5.21264e7 −1.77693
\(137\) −1.11699e7 −0.371132 −0.185566 0.982632i \(-0.559412\pi\)
−0.185566 + 0.982632i \(0.559412\pi\)
\(138\) −4.06323e6 −0.131612
\(139\) 4.22972e7 1.33586 0.667928 0.744226i \(-0.267182\pi\)
0.667928 + 0.744226i \(0.267182\pi\)
\(140\) −4.78449e6 −0.147363
\(141\) 3.35397e7 1.00761
\(142\) 1.53035e7 0.448518
\(143\) −2.88994e7 −0.826442
\(144\) −9.31059e6 −0.259842
\(145\) −7.64613e7 −2.08283
\(146\) −1.41836e7 −0.377183
\(147\) −1.53342e7 −0.398155
\(148\) −7.61027e6 −0.192967
\(149\) 5.92562e7 1.46751 0.733757 0.679412i \(-0.237765\pi\)
0.733757 + 0.679412i \(0.237765\pi\)
\(150\) 2.20556e7 0.533579
\(151\) 7.21512e7 1.70539 0.852697 0.522406i \(-0.174966\pi\)
0.852697 + 0.522406i \(0.174966\pi\)
\(152\) −6.47899e7 −1.49642
\(153\) 2.45235e7 0.553556
\(154\) 2.83417e7 0.625321
\(155\) 8.86316e7 1.91174
\(156\) −3.38121e6 −0.0713076
\(157\) −5.11820e7 −1.05552 −0.527762 0.849392i \(-0.676969\pi\)
−0.527762 + 0.849392i \(0.676969\pi\)
\(158\) 7.28421e7 1.46921
\(159\) −9.35161e6 −0.184500
\(160\) −2.70291e7 −0.521690
\(161\) 7.45339e6 0.140755
\(162\) 5.42495e6 0.100252
\(163\) 3.02638e7 0.547352 0.273676 0.961822i \(-0.411760\pi\)
0.273676 + 0.961822i \(0.411760\pi\)
\(164\) 4.99736e6 0.0884682
\(165\) 5.89649e7 1.02188
\(166\) 2.07557e7 0.352175
\(167\) 5.48855e7 0.911906 0.455953 0.890004i \(-0.349298\pi\)
0.455953 + 0.890004i \(0.349298\pi\)
\(168\) 2.11521e7 0.344169
\(169\) −3.50548e7 −0.558655
\(170\) −1.36561e8 −2.13184
\(171\) 3.04812e7 0.466171
\(172\) 4.70814e6 0.0705504
\(173\) −9.94511e7 −1.46032 −0.730161 0.683276i \(-0.760555\pi\)
−0.730161 + 0.683276i \(0.760555\pi\)
\(174\) 5.29925e7 0.762592
\(175\) −4.04577e7 −0.570647
\(176\) 7.01371e7 0.969736
\(177\) 5.54523e6 0.0751646
\(178\) 5.63637e7 0.749083
\(179\) 1.27756e8 1.66493 0.832465 0.554078i \(-0.186929\pi\)
0.832465 + 0.554078i \(0.186929\pi\)
\(180\) 6.89885e6 0.0881704
\(181\) 5.65804e7 0.709237 0.354618 0.935011i \(-0.384611\pi\)
0.354618 + 0.935011i \(0.384611\pi\)
\(182\) −2.71593e7 −0.333940
\(183\) −7.52791e7 −0.908020
\(184\) 2.28439e7 0.270338
\(185\) −1.27179e8 −1.47677
\(186\) −6.14273e7 −0.699949
\(187\) −1.84736e8 −2.06589
\(188\) −2.95605e7 −0.324459
\(189\) −9.95127e6 −0.107217
\(190\) −1.69737e8 −1.79531
\(191\) −1.10143e8 −1.14378 −0.571888 0.820331i \(-0.693789\pi\)
−0.571888 + 0.820331i \(0.693789\pi\)
\(192\) 6.28720e7 0.641066
\(193\) 8.48163e7 0.849236 0.424618 0.905373i \(-0.360408\pi\)
0.424618 + 0.905373i \(0.360408\pi\)
\(194\) 1.53804e8 1.51239
\(195\) −5.65049e7 −0.545714
\(196\) 1.35150e7 0.128209
\(197\) 8.45072e7 0.787520 0.393760 0.919213i \(-0.371174\pi\)
0.393760 + 0.919213i \(0.371174\pi\)
\(198\) −4.08664e7 −0.374143
\(199\) −1.14880e8 −1.03337 −0.516686 0.856175i \(-0.672835\pi\)
−0.516686 + 0.856175i \(0.672835\pi\)
\(200\) −1.23999e8 −1.09600
\(201\) −1.24586e8 −1.08214
\(202\) −7.25360e7 −0.619190
\(203\) −9.72070e7 −0.815570
\(204\) −2.16140e7 −0.178250
\(205\) 8.35132e7 0.677043
\(206\) 5.53129e7 0.440850
\(207\) −1.07472e7 −0.0842167
\(208\) −6.72110e7 −0.517868
\(209\) −2.29616e8 −1.73976
\(210\) 5.54145e7 0.412910
\(211\) −2.23391e8 −1.63711 −0.818555 0.574428i \(-0.805225\pi\)
−0.818555 + 0.574428i \(0.805225\pi\)
\(212\) 8.24214e6 0.0594107
\(213\) 4.04774e7 0.287001
\(214\) 2.02303e8 1.41109
\(215\) 7.86798e7 0.539919
\(216\) −3.04996e7 −0.205924
\(217\) 1.12679e8 0.748575
\(218\) 1.00075e8 0.654229
\(219\) −3.75155e7 −0.241355
\(220\) −5.19693e7 −0.329054
\(221\) 1.77029e8 1.10325
\(222\) 8.81428e7 0.540694
\(223\) −1.25281e8 −0.756519 −0.378259 0.925700i \(-0.623477\pi\)
−0.378259 + 0.925700i \(0.623477\pi\)
\(224\) −3.43627e7 −0.204277
\(225\) 5.83366e7 0.341431
\(226\) −1.26608e8 −0.729596
\(227\) −1.69633e7 −0.0962540 −0.0481270 0.998841i \(-0.515325\pi\)
−0.0481270 + 0.998841i \(0.515325\pi\)
\(228\) −2.68649e7 −0.150111
\(229\) −9.47495e7 −0.521378 −0.260689 0.965423i \(-0.583950\pi\)
−0.260689 + 0.965423i \(0.583950\pi\)
\(230\) 5.98465e7 0.324333
\(231\) 7.49633e7 0.400135
\(232\) −2.97929e8 −1.56641
\(233\) −1.23601e8 −0.640143 −0.320072 0.947393i \(-0.603707\pi\)
−0.320072 + 0.947393i \(0.603707\pi\)
\(234\) 3.91615e7 0.199804
\(235\) −4.93999e8 −2.48307
\(236\) −4.88735e6 −0.0242037
\(237\) 1.92666e8 0.940127
\(238\) −1.73613e8 −0.834762
\(239\) 1.84669e8 0.874985 0.437493 0.899222i \(-0.355867\pi\)
0.437493 + 0.899222i \(0.355867\pi\)
\(240\) 1.37134e8 0.640333
\(241\) −3.10641e8 −1.42955 −0.714775 0.699354i \(-0.753471\pi\)
−0.714775 + 0.699354i \(0.753471\pi\)
\(242\) 1.08923e8 0.494044
\(243\) 1.43489e7 0.0641500
\(244\) 6.63480e7 0.292391
\(245\) 2.25855e8 0.981181
\(246\) −5.78799e7 −0.247888
\(247\) 2.20037e8 0.929086
\(248\) 3.45351e8 1.43774
\(249\) 5.48984e7 0.225352
\(250\) −7.70404e6 −0.0311838
\(251\) 1.38239e8 0.551788 0.275894 0.961188i \(-0.411026\pi\)
0.275894 + 0.961188i \(0.411026\pi\)
\(252\) 8.77066e6 0.0345247
\(253\) 8.09589e7 0.314299
\(254\) 9.31915e7 0.356827
\(255\) −3.61202e8 −1.36414
\(256\) −1.44221e8 −0.537264
\(257\) −1.25743e8 −0.462080 −0.231040 0.972944i \(-0.574213\pi\)
−0.231040 + 0.972944i \(0.574213\pi\)
\(258\) −5.45301e7 −0.197682
\(259\) −1.61685e8 −0.578257
\(260\) 4.98012e7 0.175725
\(261\) 1.40164e8 0.487973
\(262\) 1.95529e8 0.671672
\(263\) −2.91293e8 −0.987380 −0.493690 0.869638i \(-0.664352\pi\)
−0.493690 + 0.869638i \(0.664352\pi\)
\(264\) 2.29755e8 0.768513
\(265\) 1.37738e8 0.454667
\(266\) −2.15790e8 −0.702985
\(267\) 1.49081e8 0.479329
\(268\) 1.09805e8 0.348459
\(269\) −7.18507e7 −0.225060 −0.112530 0.993648i \(-0.535895\pi\)
−0.112530 + 0.993648i \(0.535895\pi\)
\(270\) −7.99031e7 −0.247053
\(271\) 2.99889e7 0.0915310 0.0457655 0.998952i \(-0.485427\pi\)
0.0457655 + 0.998952i \(0.485427\pi\)
\(272\) −4.29639e8 −1.29453
\(273\) −7.18359e7 −0.213684
\(274\) −1.14023e8 −0.334861
\(275\) −4.39452e8 −1.27423
\(276\) 9.47213e6 0.0271185
\(277\) 3.52435e8 0.996322 0.498161 0.867085i \(-0.334009\pi\)
0.498161 + 0.867085i \(0.334009\pi\)
\(278\) 4.31769e8 1.20530
\(279\) −1.62474e8 −0.447889
\(280\) −3.11546e8 −0.848142
\(281\) 3.70901e8 0.997209 0.498605 0.866830i \(-0.333846\pi\)
0.498605 + 0.866830i \(0.333846\pi\)
\(282\) 3.42373e8 0.909133
\(283\) 1.18723e8 0.311373 0.155687 0.987807i \(-0.450241\pi\)
0.155687 + 0.987807i \(0.450241\pi\)
\(284\) −3.56752e7 −0.0924170
\(285\) −4.48952e8 −1.14880
\(286\) −2.95005e8 −0.745673
\(287\) 1.06172e8 0.265109
\(288\) 4.95482e7 0.122223
\(289\) 7.21302e8 1.75782
\(290\) −7.80517e8 −1.87927
\(291\) 4.06810e8 0.967758
\(292\) 3.30647e7 0.0777184
\(293\) 6.18045e8 1.43544 0.717718 0.696334i \(-0.245187\pi\)
0.717718 + 0.696334i \(0.245187\pi\)
\(294\) −1.56532e8 −0.359242
\(295\) −8.16747e7 −0.185230
\(296\) −4.95548e8 −1.11062
\(297\) −1.08091e8 −0.239410
\(298\) 6.04888e8 1.32409
\(299\) −7.75814e7 −0.167845
\(300\) −5.14156e7 −0.109944
\(301\) 1.00027e8 0.211415
\(302\) 7.36520e8 1.53872
\(303\) −1.91857e8 −0.396212
\(304\) −5.34016e8 −1.09018
\(305\) 1.10877e9 2.23765
\(306\) 2.50335e8 0.499456
\(307\) −7.10474e8 −1.40141 −0.700703 0.713453i \(-0.747130\pi\)
−0.700703 + 0.713453i \(0.747130\pi\)
\(308\) −6.60697e7 −0.128847
\(309\) 1.46302e8 0.282094
\(310\) 9.04752e8 1.72490
\(311\) 8.99410e8 1.69549 0.847747 0.530401i \(-0.177958\pi\)
0.847747 + 0.530401i \(0.177958\pi\)
\(312\) −2.20170e8 −0.410409
\(313\) −1.68969e8 −0.311459 −0.155729 0.987800i \(-0.549773\pi\)
−0.155729 + 0.987800i \(0.549773\pi\)
\(314\) −5.22466e8 −0.952366
\(315\) 1.46570e8 0.264216
\(316\) −1.69808e8 −0.302729
\(317\) 6.45496e8 1.13812 0.569058 0.822297i \(-0.307308\pi\)
0.569058 + 0.822297i \(0.307308\pi\)
\(318\) −9.54613e7 −0.166469
\(319\) −1.05586e9 −1.82113
\(320\) −9.26030e8 −1.57979
\(321\) 5.35088e8 0.902938
\(322\) 7.60842e7 0.126999
\(323\) 1.40656e9 2.32247
\(324\) −1.26466e7 −0.0206569
\(325\) 4.21119e8 0.680476
\(326\) 3.08933e8 0.493858
\(327\) 2.64698e8 0.418633
\(328\) 3.25407e8 0.509176
\(329\) −6.28032e8 −0.972291
\(330\) 6.01913e8 0.922009
\(331\) −1.03251e9 −1.56493 −0.782467 0.622692i \(-0.786039\pi\)
−0.782467 + 0.622692i \(0.786039\pi\)
\(332\) −4.83853e7 −0.0725655
\(333\) 2.33136e8 0.345983
\(334\) 5.60272e8 0.822784
\(335\) 1.83501e9 2.66674
\(336\) 1.74341e8 0.250734
\(337\) −2.83276e8 −0.403186 −0.201593 0.979469i \(-0.564612\pi\)
−0.201593 + 0.979469i \(0.564612\pi\)
\(338\) −3.57839e8 −0.504057
\(339\) −3.34877e8 −0.466859
\(340\) 3.18349e8 0.439265
\(341\) 1.22393e9 1.67153
\(342\) 3.11152e8 0.420611
\(343\) 7.03499e8 0.941313
\(344\) 3.06574e8 0.406051
\(345\) 1.58293e8 0.207537
\(346\) −1.01520e9 −1.31760
\(347\) 3.08826e8 0.396790 0.198395 0.980122i \(-0.436427\pi\)
0.198395 + 0.980122i \(0.436427\pi\)
\(348\) −1.23535e8 −0.157132
\(349\) −9.76048e8 −1.22909 −0.614543 0.788884i \(-0.710659\pi\)
−0.614543 + 0.788884i \(0.710659\pi\)
\(350\) −4.12992e8 −0.514877
\(351\) 1.03581e8 0.127852
\(352\) −3.73249e8 −0.456141
\(353\) −6.80666e8 −0.823612 −0.411806 0.911271i \(-0.635102\pi\)
−0.411806 + 0.911271i \(0.635102\pi\)
\(354\) 5.66057e7 0.0678186
\(355\) −5.96184e8 −0.707263
\(356\) −1.31394e8 −0.154348
\(357\) −4.59203e8 −0.534154
\(358\) 1.30413e9 1.50221
\(359\) 8.85168e8 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(360\) 4.49223e8 0.507462
\(361\) 8.54398e8 0.955840
\(362\) 5.77573e8 0.639922
\(363\) 2.88100e8 0.316133
\(364\) 6.33134e7 0.0688083
\(365\) 5.52558e8 0.594776
\(366\) −7.68449e8 −0.819277
\(367\) 1.33637e9 1.41122 0.705610 0.708601i \(-0.250673\pi\)
0.705610 + 0.708601i \(0.250673\pi\)
\(368\) 1.88285e8 0.196947
\(369\) −1.53091e8 −0.158620
\(370\) −1.29824e9 −1.33244
\(371\) 1.75110e8 0.178033
\(372\) 1.43198e8 0.144224
\(373\) 6.67780e8 0.666274 0.333137 0.942879i \(-0.391893\pi\)
0.333137 + 0.942879i \(0.391893\pi\)
\(374\) −1.88579e9 −1.86398
\(375\) −2.03771e7 −0.0199541
\(376\) −1.92485e9 −1.86741
\(377\) 1.01182e9 0.972538
\(378\) −1.01583e8 −0.0967382
\(379\) 1.22986e9 1.16043 0.580213 0.814465i \(-0.302969\pi\)
0.580213 + 0.814465i \(0.302969\pi\)
\(380\) 3.95688e8 0.369922
\(381\) 2.46490e8 0.228329
\(382\) −1.12434e9 −1.03199
\(383\) 1.72139e9 1.56561 0.782805 0.622267i \(-0.213788\pi\)
0.782805 + 0.622267i \(0.213788\pi\)
\(384\) 4.06902e8 0.366717
\(385\) −1.10412e9 −0.986062
\(386\) 8.65804e8 0.766239
\(387\) −1.44231e8 −0.126494
\(388\) −3.58547e8 −0.311627
\(389\) −8.54144e8 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(390\) −5.76802e8 −0.492380
\(391\) −4.95931e8 −0.419568
\(392\) 8.80039e8 0.737905
\(393\) 5.17172e8 0.429795
\(394\) 8.62649e8 0.710555
\(395\) −2.83775e9 −2.31677
\(396\) 9.52671e7 0.0770921
\(397\) 4.64867e8 0.372874 0.186437 0.982467i \(-0.440306\pi\)
0.186437 + 0.982467i \(0.440306\pi\)
\(398\) −1.17269e9 −0.932379
\(399\) −5.70762e8 −0.449832
\(400\) −1.02203e9 −0.798461
\(401\) 1.35635e9 1.05043 0.525216 0.850969i \(-0.323985\pi\)
0.525216 + 0.850969i \(0.323985\pi\)
\(402\) −1.27178e9 −0.976382
\(403\) −1.17287e9 −0.892648
\(404\) 1.69095e8 0.127584
\(405\) −2.11342e8 −0.158086
\(406\) −9.92289e8 −0.735863
\(407\) −1.75623e9 −1.29122
\(408\) −1.40741e9 −1.02591
\(409\) −1.50649e9 −1.08876 −0.544382 0.838838i \(-0.683236\pi\)
−0.544382 + 0.838838i \(0.683236\pi\)
\(410\) 8.52502e8 0.610875
\(411\) −3.01588e8 −0.214273
\(412\) −1.28944e8 −0.0908370
\(413\) −1.03835e8 −0.0725301
\(414\) −1.09707e8 −0.0759860
\(415\) −8.08589e8 −0.555341
\(416\) 3.57677e8 0.243593
\(417\) 1.14202e9 0.771256
\(418\) −2.34392e9 −1.56973
\(419\) −2.73741e9 −1.81799 −0.908995 0.416808i \(-0.863149\pi\)
−0.908995 + 0.416808i \(0.863149\pi\)
\(420\) −1.29181e8 −0.0850800
\(421\) −4.08418e8 −0.266758 −0.133379 0.991065i \(-0.542583\pi\)
−0.133379 + 0.991065i \(0.542583\pi\)
\(422\) −2.28038e9 −1.47711
\(423\) 9.05571e8 0.581743
\(424\) 5.36693e8 0.341936
\(425\) 2.69196e9 1.70101
\(426\) 4.13193e8 0.258952
\(427\) 1.40961e9 0.876193
\(428\) −4.71606e8 −0.290754
\(429\) −7.80284e8 −0.477147
\(430\) 8.03164e8 0.487152
\(431\) −7.95781e8 −0.478766 −0.239383 0.970925i \(-0.576945\pi\)
−0.239383 + 0.970925i \(0.576945\pi\)
\(432\) −2.51386e8 −0.150020
\(433\) −1.70983e9 −1.01215 −0.506076 0.862489i \(-0.668905\pi\)
−0.506076 + 0.862489i \(0.668905\pi\)
\(434\) 1.15023e9 0.675415
\(435\) −2.06446e9 −1.20252
\(436\) −2.33294e8 −0.134804
\(437\) −6.16412e8 −0.353334
\(438\) −3.82958e8 −0.217767
\(439\) 3.43402e9 1.93721 0.968606 0.248602i \(-0.0799712\pi\)
0.968606 + 0.248602i \(0.0799712\pi\)
\(440\) −3.38402e9 −1.89386
\(441\) −4.14025e8 −0.229875
\(442\) 1.80711e9 0.995423
\(443\) −8.23020e7 −0.0449777 −0.0224888 0.999747i \(-0.507159\pi\)
−0.0224888 + 0.999747i \(0.507159\pi\)
\(444\) −2.05477e8 −0.111410
\(445\) −2.19579e9 −1.18122
\(446\) −1.27887e9 −0.682583
\(447\) 1.59992e9 0.847269
\(448\) −1.17728e9 −0.618597
\(449\) −1.42868e9 −0.744856 −0.372428 0.928061i \(-0.621475\pi\)
−0.372428 + 0.928061i \(0.621475\pi\)
\(450\) 5.95500e8 0.308062
\(451\) 1.15324e9 0.591975
\(452\) 2.95147e8 0.150333
\(453\) 1.94808e9 0.984609
\(454\) −1.73161e8 −0.0868469
\(455\) 1.05806e9 0.526586
\(456\) −1.74933e9 −0.863961
\(457\) 1.78047e9 0.872625 0.436312 0.899795i \(-0.356284\pi\)
0.436312 + 0.899795i \(0.356284\pi\)
\(458\) −9.67203e8 −0.470423
\(459\) 6.62133e8 0.319596
\(460\) −1.39513e8 −0.0668287
\(461\) 2.67034e9 1.26944 0.634722 0.772741i \(-0.281115\pi\)
0.634722 + 0.772741i \(0.281115\pi\)
\(462\) 7.65226e8 0.361029
\(463\) 9.93730e7 0.0465302 0.0232651 0.999729i \(-0.492594\pi\)
0.0232651 + 0.999729i \(0.492594\pi\)
\(464\) −2.45561e9 −1.14116
\(465\) 2.39305e9 1.10374
\(466\) −1.26172e9 −0.577581
\(467\) −2.09963e9 −0.953967 −0.476984 0.878912i \(-0.658270\pi\)
−0.476984 + 0.878912i \(0.658270\pi\)
\(468\) −9.12926e7 −0.0411695
\(469\) 2.33289e9 1.04421
\(470\) −5.04274e9 −2.24039
\(471\) −1.38191e9 −0.609407
\(472\) −3.18243e8 −0.139304
\(473\) 1.08650e9 0.472080
\(474\) 1.96674e9 0.848247
\(475\) 3.34594e9 1.43249
\(476\) 4.04724e8 0.172002
\(477\) −2.52494e8 −0.106521
\(478\) 1.88510e9 0.789471
\(479\) 3.04269e8 0.126498 0.0632490 0.997998i \(-0.479854\pi\)
0.0632490 + 0.997998i \(0.479854\pi\)
\(480\) −7.29787e8 −0.301198
\(481\) 1.68296e9 0.689550
\(482\) −3.17103e9 −1.28984
\(483\) 2.01242e8 0.0812648
\(484\) −2.53920e8 −0.101798
\(485\) −5.99184e9 −2.38487
\(486\) 1.46474e8 0.0578805
\(487\) −8.62328e8 −0.338315 −0.169157 0.985589i \(-0.554105\pi\)
−0.169157 + 0.985589i \(0.554105\pi\)
\(488\) 4.32030e9 1.68284
\(489\) 8.17122e8 0.316014
\(490\) 2.30553e9 0.885288
\(491\) −1.09588e9 −0.417811 −0.208905 0.977936i \(-0.566990\pi\)
−0.208905 + 0.977936i \(0.566990\pi\)
\(492\) 1.34929e8 0.0510771
\(493\) 6.46792e9 2.43109
\(494\) 2.24613e9 0.838284
\(495\) 1.59205e9 0.589982
\(496\) 2.84647e9 1.04742
\(497\) −7.57942e8 −0.276942
\(498\) 5.60403e8 0.203328
\(499\) 2.67449e9 0.963582 0.481791 0.876286i \(-0.339986\pi\)
0.481791 + 0.876286i \(0.339986\pi\)
\(500\) 1.79595e7 0.00642540
\(501\) 1.48191e9 0.526489
\(502\) 1.41114e9 0.497861
\(503\) −4.00214e9 −1.40218 −0.701091 0.713072i \(-0.747303\pi\)
−0.701091 + 0.713072i \(0.747303\pi\)
\(504\) 5.71107e8 0.198706
\(505\) 2.82582e9 0.976394
\(506\) 8.26428e8 0.283582
\(507\) −9.46479e8 −0.322540
\(508\) −2.17247e8 −0.0735241
\(509\) 1.64260e9 0.552102 0.276051 0.961143i \(-0.410974\pi\)
0.276051 + 0.961143i \(0.410974\pi\)
\(510\) −3.68715e9 −1.23082
\(511\) 7.02480e8 0.232895
\(512\) −3.40122e9 −1.11993
\(513\) 8.22992e8 0.269144
\(514\) −1.28358e9 −0.416920
\(515\) −2.15485e9 −0.695171
\(516\) 1.27120e8 0.0407323
\(517\) −6.82170e9 −2.17108
\(518\) −1.65048e9 −0.521743
\(519\) −2.68518e9 −0.843117
\(520\) 3.24284e9 1.01138
\(521\) 7.57722e8 0.234735 0.117368 0.993089i \(-0.462554\pi\)
0.117368 + 0.993089i \(0.462554\pi\)
\(522\) 1.43080e9 0.440283
\(523\) 7.00512e8 0.214121 0.107061 0.994252i \(-0.465856\pi\)
0.107061 + 0.994252i \(0.465856\pi\)
\(524\) −4.55815e8 −0.138398
\(525\) −1.09236e9 −0.329463
\(526\) −2.97352e9 −0.890882
\(527\) −7.49741e9 −2.23138
\(528\) 1.89370e9 0.559877
\(529\) −3.18749e9 −0.936168
\(530\) 1.40603e9 0.410232
\(531\) 1.49721e8 0.0433963
\(532\) 5.03047e8 0.144850
\(533\) −1.10513e9 −0.316132
\(534\) 1.52182e9 0.432483
\(535\) −7.88122e9 −2.22513
\(536\) 7.15006e9 2.00555
\(537\) 3.44941e9 0.961248
\(538\) −7.33452e8 −0.203064
\(539\) 3.11887e9 0.857899
\(540\) 1.86269e8 0.0509052
\(541\) −1.39710e9 −0.379346 −0.189673 0.981847i \(-0.560743\pi\)
−0.189673 + 0.981847i \(0.560743\pi\)
\(542\) 3.06127e8 0.0825855
\(543\) 1.52767e9 0.409478
\(544\) 2.28641e9 0.608918
\(545\) −3.89869e9 −1.03165
\(546\) −7.33301e8 −0.192801
\(547\) −1.96444e8 −0.0513197 −0.0256598 0.999671i \(-0.508169\pi\)
−0.0256598 + 0.999671i \(0.508169\pi\)
\(548\) 2.65808e8 0.0689979
\(549\) −2.03253e9 −0.524245
\(550\) −4.48593e9 −1.14970
\(551\) 8.03923e9 2.04731
\(552\) 6.16785e8 0.156080
\(553\) −3.60769e9 −0.907175
\(554\) 3.59766e9 0.898950
\(555\) −3.43382e9 −0.852614
\(556\) −1.00653e9 −0.248352
\(557\) 2.95103e9 0.723571 0.361785 0.932261i \(-0.382167\pi\)
0.361785 + 0.932261i \(0.382167\pi\)
\(558\) −1.65854e9 −0.404116
\(559\) −1.04117e9 −0.252105
\(560\) −2.56784e9 −0.617889
\(561\) −4.98788e9 −1.19274
\(562\) 3.78616e9 0.899750
\(563\) −9.45756e7 −0.0223357 −0.0111679 0.999938i \(-0.503555\pi\)
−0.0111679 + 0.999938i \(0.503555\pi\)
\(564\) −7.98134e8 −0.187326
\(565\) 4.93234e9 1.15049
\(566\) 1.21192e9 0.280942
\(567\) −2.68684e8 −0.0619015
\(568\) −2.32301e9 −0.531903
\(569\) 6.34844e8 0.144469 0.0722344 0.997388i \(-0.476987\pi\)
0.0722344 + 0.997388i \(0.476987\pi\)
\(570\) −4.58290e9 −1.03652
\(571\) 6.22435e9 1.39916 0.699580 0.714554i \(-0.253370\pi\)
0.699580 + 0.714554i \(0.253370\pi\)
\(572\) 6.87711e8 0.153646
\(573\) −2.97387e9 −0.660360
\(574\) 1.08381e9 0.239199
\(575\) −1.17972e9 −0.258787
\(576\) 1.69754e9 0.370120
\(577\) 5.59231e9 1.21193 0.605963 0.795493i \(-0.292788\pi\)
0.605963 + 0.795493i \(0.292788\pi\)
\(578\) 7.36305e9 1.58603
\(579\) 2.29004e9 0.490307
\(580\) 1.81953e9 0.387223
\(581\) −1.02798e9 −0.217454
\(582\) 4.15272e9 0.873177
\(583\) 1.90204e9 0.397540
\(584\) 2.15303e9 0.447306
\(585\) −1.52563e9 −0.315068
\(586\) 6.30901e9 1.29515
\(587\) 5.92527e9 1.20913 0.604567 0.796554i \(-0.293346\pi\)
0.604567 + 0.796554i \(0.293346\pi\)
\(588\) 3.64905e8 0.0740217
\(589\) −9.31883e9 −1.87913
\(590\) −8.33735e8 −0.167127
\(591\) 2.28169e9 0.454675
\(592\) −4.08444e9 −0.809108
\(593\) −4.66790e9 −0.919242 −0.459621 0.888115i \(-0.652015\pi\)
−0.459621 + 0.888115i \(0.652015\pi\)
\(594\) −1.10339e9 −0.216012
\(595\) 6.76352e9 1.31633
\(596\) −1.41011e9 −0.272828
\(597\) −3.10175e9 −0.596618
\(598\) −7.91951e8 −0.151441
\(599\) 2.89602e9 0.550563 0.275282 0.961364i \(-0.411229\pi\)
0.275282 + 0.961364i \(0.411229\pi\)
\(600\) −3.34796e9 −0.632778
\(601\) 4.95609e9 0.931276 0.465638 0.884975i \(-0.345825\pi\)
0.465638 + 0.884975i \(0.345825\pi\)
\(602\) 1.02108e9 0.190753
\(603\) −3.36383e9 −0.624774
\(604\) −1.71696e9 −0.317053
\(605\) −4.24337e9 −0.779053
\(606\) −1.95847e9 −0.357490
\(607\) −2.16921e9 −0.393678 −0.196839 0.980436i \(-0.563068\pi\)
−0.196839 + 0.980436i \(0.563068\pi\)
\(608\) 2.84187e9 0.512793
\(609\) −2.62459e9 −0.470870
\(610\) 1.13183e10 2.01896
\(611\) 6.53711e9 1.15942
\(612\) −5.83578e8 −0.102913
\(613\) −2.26724e9 −0.397544 −0.198772 0.980046i \(-0.563695\pi\)
−0.198772 + 0.980046i \(0.563695\pi\)
\(614\) −7.25252e9 −1.26444
\(615\) 2.25486e9 0.390891
\(616\) −4.30218e9 −0.741576
\(617\) 3.64399e9 0.624567 0.312284 0.949989i \(-0.398906\pi\)
0.312284 + 0.949989i \(0.398906\pi\)
\(618\) 1.49345e9 0.254525
\(619\) 8.80200e9 1.49164 0.745820 0.666147i \(-0.232058\pi\)
0.745820 + 0.666147i \(0.232058\pi\)
\(620\) −2.10914e9 −0.355415
\(621\) −2.90174e8 −0.0486225
\(622\) 9.18118e9 1.52979
\(623\) −2.79155e9 −0.462528
\(624\) −1.81470e9 −0.298991
\(625\) −5.95165e9 −0.975118
\(626\) −1.72483e9 −0.281019
\(627\) −6.19963e9 −1.00445
\(628\) 1.21796e9 0.196235
\(629\) 1.07581e10 1.72369
\(630\) 1.49619e9 0.238394
\(631\) 3.58813e9 0.568546 0.284273 0.958743i \(-0.408248\pi\)
0.284273 + 0.958743i \(0.408248\pi\)
\(632\) −1.10572e10 −1.74235
\(633\) −6.03157e9 −0.945186
\(634\) 6.58923e9 1.02689
\(635\) −3.63051e9 −0.562677
\(636\) 2.22538e8 0.0343008
\(637\) −2.98875e9 −0.458143
\(638\) −1.07783e10 −1.64315
\(639\) 1.09289e9 0.165700
\(640\) −5.99319e9 −0.903708
\(641\) 5.09433e9 0.763984 0.381992 0.924166i \(-0.375238\pi\)
0.381992 + 0.924166i \(0.375238\pi\)
\(642\) 5.46218e9 0.814692
\(643\) −6.13203e9 −0.909633 −0.454816 0.890585i \(-0.650295\pi\)
−0.454816 + 0.890585i \(0.650295\pi\)
\(644\) −1.77366e8 −0.0261680
\(645\) 2.12436e9 0.311723
\(646\) 1.43582e10 2.09549
\(647\) −7.78100e9 −1.12946 −0.564729 0.825276i \(-0.691019\pi\)
−0.564729 + 0.825276i \(0.691019\pi\)
\(648\) −8.23490e8 −0.118890
\(649\) −1.12786e9 −0.161956
\(650\) 4.29878e9 0.613972
\(651\) 3.04234e9 0.432190
\(652\) −7.20179e8 −0.101759
\(653\) −6.82855e8 −0.0959693 −0.0479846 0.998848i \(-0.515280\pi\)
−0.0479846 + 0.998848i \(0.515280\pi\)
\(654\) 2.70204e9 0.377719
\(655\) −7.61733e9 −1.05915
\(656\) 2.68209e9 0.370945
\(657\) −1.01292e9 −0.139346
\(658\) −6.41095e9 −0.877267
\(659\) 6.06750e8 0.0825868 0.0412934 0.999147i \(-0.486852\pi\)
0.0412934 + 0.999147i \(0.486852\pi\)
\(660\) −1.40317e9 −0.189980
\(661\) −1.22400e9 −0.164846 −0.0824228 0.996597i \(-0.526266\pi\)
−0.0824228 + 0.996597i \(0.526266\pi\)
\(662\) −1.05399e10 −1.41199
\(663\) 4.77979e9 0.636959
\(664\) −3.15064e9 −0.417649
\(665\) 8.40665e9 1.10853
\(666\) 2.37986e9 0.312170
\(667\) −2.83450e9 −0.369859
\(668\) −1.30610e9 −0.169534
\(669\) −3.38260e9 −0.436776
\(670\) 1.87318e10 2.40612
\(671\) 1.53112e10 1.95650
\(672\) −9.27794e8 −0.117939
\(673\) −8.76801e9 −1.10879 −0.554394 0.832254i \(-0.687050\pi\)
−0.554394 + 0.832254i \(0.687050\pi\)
\(674\) −2.89168e9 −0.363782
\(675\) 1.57509e9 0.197125
\(676\) 8.34189e8 0.103861
\(677\) 3.66670e9 0.454166 0.227083 0.973875i \(-0.427081\pi\)
0.227083 + 0.973875i \(0.427081\pi\)
\(678\) −3.41842e9 −0.421232
\(679\) −7.61755e9 −0.933837
\(680\) 2.07295e10 2.52818
\(681\) −4.58008e8 −0.0555723
\(682\) 1.24938e10 1.50817
\(683\) −2.41403e8 −0.0289915 −0.0144957 0.999895i \(-0.504614\pi\)
−0.0144957 + 0.999895i \(0.504614\pi\)
\(684\) −7.25353e8 −0.0866668
\(685\) 4.44203e9 0.528038
\(686\) 7.18132e9 0.849317
\(687\) −2.55824e9 −0.301018
\(688\) 2.52686e9 0.295816
\(689\) −1.82269e9 −0.212298
\(690\) 1.61586e9 0.187254
\(691\) 5.87544e9 0.677434 0.338717 0.940888i \(-0.390007\pi\)
0.338717 + 0.940888i \(0.390007\pi\)
\(692\) 2.36661e9 0.271491
\(693\) 2.02401e9 0.231018
\(694\) 3.15249e9 0.358011
\(695\) −1.68206e10 −1.90062
\(696\) −8.04409e9 −0.904367
\(697\) −7.06444e9 −0.790247
\(698\) −9.96349e9 −1.10896
\(699\) −3.33723e9 −0.369587
\(700\) 9.62760e8 0.106090
\(701\) −1.02557e10 −1.12449 −0.562243 0.826972i \(-0.690061\pi\)
−0.562243 + 0.826972i \(0.690061\pi\)
\(702\) 1.05736e9 0.115357
\(703\) 1.33717e10 1.45159
\(704\) −1.27877e10 −1.38130
\(705\) −1.33380e10 −1.43360
\(706\) −6.94824e9 −0.743119
\(707\) 3.59253e9 0.382325
\(708\) −1.31958e8 −0.0139740
\(709\) −5.18318e9 −0.546178 −0.273089 0.961989i \(-0.588045\pi\)
−0.273089 + 0.961989i \(0.588045\pi\)
\(710\) −6.08585e9 −0.638141
\(711\) 5.20199e9 0.542783
\(712\) −8.55583e9 −0.888346
\(713\) 3.28567e9 0.339477
\(714\) −4.68755e9 −0.481950
\(715\) 1.14927e10 1.17584
\(716\) −3.04018e9 −0.309530
\(717\) 4.98606e9 0.505173
\(718\) 9.03579e9 0.911026
\(719\) 1.29607e10 1.30040 0.650202 0.759762i \(-0.274684\pi\)
0.650202 + 0.759762i \(0.274684\pi\)
\(720\) 3.70262e9 0.369696
\(721\) −2.73951e9 −0.272207
\(722\) 8.72170e9 0.862424
\(723\) −8.38731e9 −0.825351
\(724\) −1.34643e9 −0.131856
\(725\) 1.53859e10 1.49948
\(726\) 2.94092e9 0.285237
\(727\) 4.17261e9 0.402752 0.201376 0.979514i \(-0.435459\pi\)
0.201376 + 0.979514i \(0.435459\pi\)
\(728\) 4.12269e9 0.396024
\(729\) 3.87420e8 0.0370370
\(730\) 5.64052e9 0.536647
\(731\) −6.65558e9 −0.630195
\(732\) 1.79140e9 0.168812
\(733\) −1.52037e10 −1.42589 −0.712944 0.701221i \(-0.752638\pi\)
−0.712944 + 0.701221i \(0.752638\pi\)
\(734\) 1.36416e10 1.27330
\(735\) 6.09809e9 0.566485
\(736\) −1.00200e9 −0.0926392
\(737\) 2.53399e10 2.33168
\(738\) −1.56276e9 −0.143118
\(739\) −7.33278e9 −0.668364 −0.334182 0.942509i \(-0.608460\pi\)
−0.334182 + 0.942509i \(0.608460\pi\)
\(740\) 3.02644e9 0.274549
\(741\) 5.94099e9 0.536408
\(742\) 1.78752e9 0.160634
\(743\) 6.55520e9 0.586307 0.293154 0.956065i \(-0.405295\pi\)
0.293154 + 0.956065i \(0.405295\pi\)
\(744\) 9.32447e9 0.830078
\(745\) −2.35649e10 −2.08794
\(746\) 6.81670e9 0.601158
\(747\) 1.48226e9 0.130107
\(748\) 4.39612e9 0.384073
\(749\) −1.00196e10 −0.871290
\(750\) −2.08009e8 −0.0180040
\(751\) −7.42395e9 −0.639581 −0.319790 0.947488i \(-0.603612\pi\)
−0.319790 + 0.947488i \(0.603612\pi\)
\(752\) −1.58652e10 −1.36045
\(753\) 3.73245e9 0.318575
\(754\) 1.03286e10 0.877490
\(755\) −2.86930e10 −2.42639
\(756\) 2.36808e8 0.0199329
\(757\) −1.49414e10 −1.25186 −0.625928 0.779880i \(-0.715280\pi\)
−0.625928 + 0.779880i \(0.715280\pi\)
\(758\) 1.25544e10 1.04701
\(759\) 2.18589e9 0.181461
\(760\) 2.57655e10 2.12908
\(761\) −1.92886e10 −1.58655 −0.793276 0.608862i \(-0.791626\pi\)
−0.793276 + 0.608862i \(0.791626\pi\)
\(762\) 2.51617e9 0.206014
\(763\) −4.95649e9 −0.403960
\(764\) 2.62105e9 0.212642
\(765\) −9.75244e9 −0.787587
\(766\) 1.75720e10 1.41260
\(767\) 1.08080e9 0.0864894
\(768\) −3.89396e9 −0.310190
\(769\) 1.59930e10 1.26820 0.634099 0.773252i \(-0.281371\pi\)
0.634099 + 0.773252i \(0.281371\pi\)
\(770\) −1.12709e10 −0.889692
\(771\) −3.39506e9 −0.266782
\(772\) −2.01835e9 −0.157883
\(773\) 2.25204e10 1.75367 0.876833 0.480794i \(-0.159652\pi\)
0.876833 + 0.480794i \(0.159652\pi\)
\(774\) −1.47231e9 −0.114132
\(775\) −1.78349e10 −1.37631
\(776\) −2.33470e10 −1.79356
\(777\) −4.36550e9 −0.333857
\(778\) −8.71910e9 −0.663809
\(779\) −8.78067e9 −0.665498
\(780\) 1.34463e9 0.101455
\(781\) −8.23278e9 −0.618398
\(782\) −5.06246e9 −0.378563
\(783\) 3.78444e9 0.281732
\(784\) 7.25352e9 0.537579
\(785\) 2.03539e10 1.50178
\(786\) 5.27929e9 0.387790
\(787\) −1.72331e10 −1.26024 −0.630119 0.776498i \(-0.716994\pi\)
−0.630119 + 0.776498i \(0.716994\pi\)
\(788\) −2.01099e9 −0.146409
\(789\) −7.86490e9 −0.570064
\(790\) −2.89677e10 −2.09035
\(791\) 6.27059e9 0.450496
\(792\) 6.20338e9 0.443701
\(793\) −1.46724e10 −1.04483
\(794\) 4.74537e9 0.336432
\(795\) 3.71893e9 0.262502
\(796\) 2.73376e9 0.192116
\(797\) −1.26332e10 −0.883911 −0.441955 0.897037i \(-0.645715\pi\)
−0.441955 + 0.897037i \(0.645715\pi\)
\(798\) −5.82634e9 −0.405869
\(799\) 4.17878e10 2.89825
\(800\) 5.43894e9 0.375577
\(801\) 4.02519e9 0.276741
\(802\) 1.38457e10 0.947771
\(803\) 7.63035e9 0.520044
\(804\) 2.96475e9 0.201183
\(805\) −2.96405e9 −0.200263
\(806\) −1.19726e10 −0.805408
\(807\) −1.93997e9 −0.129938
\(808\) 1.10107e10 0.734305
\(809\) 9.80378e9 0.650989 0.325495 0.945544i \(-0.394469\pi\)
0.325495 + 0.945544i \(0.394469\pi\)
\(810\) −2.15738e9 −0.142636
\(811\) −1.10475e10 −0.727262 −0.363631 0.931543i \(-0.618463\pi\)
−0.363631 + 0.931543i \(0.618463\pi\)
\(812\) 2.31321e9 0.151624
\(813\) 8.09701e8 0.0528455
\(814\) −1.79276e10 −1.16503
\(815\) −1.20352e10 −0.778759
\(816\) −1.16003e10 −0.747399
\(817\) −8.27249e9 −0.530712
\(818\) −1.53782e10 −0.982357
\(819\) −1.93957e9 −0.123371
\(820\) −1.98734e9 −0.125870
\(821\) −1.50386e10 −0.948433 −0.474217 0.880408i \(-0.657269\pi\)
−0.474217 + 0.880408i \(0.657269\pi\)
\(822\) −3.07861e9 −0.193332
\(823\) −1.10764e10 −0.692625 −0.346313 0.938119i \(-0.612566\pi\)
−0.346313 + 0.938119i \(0.612566\pi\)
\(824\) −8.39631e9 −0.522809
\(825\) −1.18652e10 −0.735676
\(826\) −1.05995e9 −0.0654416
\(827\) 3.12849e10 1.92338 0.961691 0.274137i \(-0.0883922\pi\)
0.961691 + 0.274137i \(0.0883922\pi\)
\(828\) 2.55748e8 0.0156569
\(829\) 1.84399e10 1.12413 0.562067 0.827092i \(-0.310006\pi\)
0.562067 + 0.827092i \(0.310006\pi\)
\(830\) −8.25408e9 −0.501066
\(831\) 9.51574e9 0.575227
\(832\) 1.22542e10 0.737654
\(833\) −1.91053e10 −1.14524
\(834\) 1.16578e10 0.695880
\(835\) −2.18268e10 −1.29744
\(836\) 5.46411e9 0.323443
\(837\) −4.38681e9 −0.258589
\(838\) −2.79435e10 −1.64031
\(839\) 2.82373e10 1.65065 0.825327 0.564654i \(-0.190991\pi\)
0.825327 + 0.564654i \(0.190991\pi\)
\(840\) −8.41173e9 −0.489675
\(841\) 1.97177e10 1.14306
\(842\) −4.16913e9 −0.240687
\(843\) 1.00143e10 0.575739
\(844\) 5.31599e9 0.304358
\(845\) 1.39405e10 0.794841
\(846\) 9.24406e9 0.524888
\(847\) −5.39469e9 −0.305052
\(848\) 4.42357e9 0.249108
\(849\) 3.20551e9 0.179771
\(850\) 2.74795e10 1.53477
\(851\) −4.71465e9 −0.262238
\(852\) −9.63230e8 −0.0533570
\(853\) −1.66862e10 −0.920523 −0.460261 0.887783i \(-0.652244\pi\)
−0.460261 + 0.887783i \(0.652244\pi\)
\(854\) 1.43893e10 0.790561
\(855\) −1.21217e10 −0.663257
\(856\) −3.07089e10 −1.67343
\(857\) −2.12716e10 −1.15443 −0.577215 0.816592i \(-0.695861\pi\)
−0.577215 + 0.816592i \(0.695861\pi\)
\(858\) −7.96514e9 −0.430514
\(859\) 2.18013e10 1.17356 0.586781 0.809745i \(-0.300395\pi\)
0.586781 + 0.809745i \(0.300395\pi\)
\(860\) −1.87232e9 −0.100377
\(861\) 2.86665e9 0.153061
\(862\) −8.12333e9 −0.431975
\(863\) −2.57691e10 −1.36478 −0.682388 0.730990i \(-0.739058\pi\)
−0.682388 + 0.730990i \(0.739058\pi\)
\(864\) 1.33780e9 0.0705657
\(865\) 3.95495e10 2.07771
\(866\) −1.74540e10 −0.913233
\(867\) 1.94751e10 1.01488
\(868\) −2.68140e9 −0.139169
\(869\) −3.91868e10 −2.02568
\(870\) −2.10740e10 −1.08500
\(871\) −2.42827e10 −1.24518
\(872\) −1.51911e10 −0.775858
\(873\) 1.09839e10 0.558735
\(874\) −6.29233e9 −0.318802
\(875\) 3.81562e8 0.0192547
\(876\) 8.92746e8 0.0448708
\(877\) −2.96656e10 −1.48510 −0.742549 0.669792i \(-0.766383\pi\)
−0.742549 + 0.669792i \(0.766383\pi\)
\(878\) 3.50545e10 1.74788
\(879\) 1.66872e10 0.828749
\(880\) −2.78920e10 −1.37972
\(881\) −2.10977e10 −1.03949 −0.519745 0.854322i \(-0.673973\pi\)
−0.519745 + 0.854322i \(0.673973\pi\)
\(882\) −4.22636e9 −0.207409
\(883\) 2.21889e9 0.108461 0.0542304 0.998528i \(-0.482729\pi\)
0.0542304 + 0.998528i \(0.482729\pi\)
\(884\) −4.21272e9 −0.205107
\(885\) −2.20522e9 −0.106942
\(886\) −8.40138e8 −0.0405819
\(887\) −7.16266e9 −0.344621 −0.172311 0.985043i \(-0.555123\pi\)
−0.172311 + 0.985043i \(0.555123\pi\)
\(888\) −1.33798e10 −0.641216
\(889\) −4.61554e9 −0.220326
\(890\) −2.24146e10 −1.06578
\(891\) −2.91845e9 −0.138223
\(892\) 2.98129e9 0.140646
\(893\) 5.19397e10 2.44073
\(894\) 1.63320e10 0.764464
\(895\) −5.08058e10 −2.36882
\(896\) −7.61927e9 −0.353863
\(897\) −2.09470e9 −0.0969053
\(898\) −1.45840e10 −0.672060
\(899\) −4.28516e10 −1.96702
\(900\) −1.38822e9 −0.0634760
\(901\) −1.16514e10 −0.530689
\(902\) 1.17723e10 0.534120
\(903\) 2.70074e9 0.122061
\(904\) 1.92187e10 0.865236
\(905\) −2.25008e10 −1.00908
\(906\) 1.98860e10 0.888382
\(907\) −8.53267e9 −0.379717 −0.189858 0.981811i \(-0.560803\pi\)
−0.189858 + 0.981811i \(0.560803\pi\)
\(908\) 4.03670e8 0.0178948
\(909\) −5.18013e9 −0.228753
\(910\) 1.08007e10 0.475122
\(911\) 2.76004e9 0.120949 0.0604744 0.998170i \(-0.480739\pi\)
0.0604744 + 0.998170i \(0.480739\pi\)
\(912\) −1.44184e10 −0.629413
\(913\) −1.11659e10 −0.485564
\(914\) 1.81750e10 0.787342
\(915\) 2.99368e10 1.29191
\(916\) 2.25473e9 0.0969304
\(917\) −9.68408e9 −0.414730
\(918\) 6.75906e9 0.288361
\(919\) 1.13500e10 0.482382 0.241191 0.970478i \(-0.422462\pi\)
0.241191 + 0.970478i \(0.422462\pi\)
\(920\) −9.08451e9 −0.384631
\(921\) −1.91828e10 −0.809102
\(922\) 2.72588e10 1.14538
\(923\) 7.88932e9 0.330243
\(924\) −1.78388e9 −0.0743900
\(925\) 2.55915e10 1.06316
\(926\) 1.01440e9 0.0419827
\(927\) 3.95014e9 0.162867
\(928\) 1.30681e10 0.536776
\(929\) 4.41532e10 1.80679 0.903395 0.428810i \(-0.141067\pi\)
0.903395 + 0.428810i \(0.141067\pi\)
\(930\) 2.44283e10 0.995870
\(931\) −2.37467e10 −0.964449
\(932\) 2.94131e9 0.119010
\(933\) 2.42841e10 0.978894
\(934\) −2.14330e10 −0.860734
\(935\) 7.34655e10 2.93929
\(936\) −5.94458e9 −0.236950
\(937\) −3.84649e10 −1.52748 −0.763741 0.645522i \(-0.776640\pi\)
−0.763741 + 0.645522i \(0.776640\pi\)
\(938\) 2.38141e10 0.942159
\(939\) −4.56215e9 −0.179821
\(940\) 1.17556e10 0.461632
\(941\) 2.95756e10 1.15710 0.578549 0.815648i \(-0.303619\pi\)
0.578549 + 0.815648i \(0.303619\pi\)
\(942\) −1.41066e10 −0.549849
\(943\) 3.09592e9 0.120226
\(944\) −2.62305e9 −0.101485
\(945\) 3.95740e9 0.152545
\(946\) 1.10910e10 0.425943
\(947\) −4.30437e10 −1.64697 −0.823483 0.567341i \(-0.807972\pi\)
−0.823483 + 0.567341i \(0.807972\pi\)
\(948\) −4.58483e9 −0.174781
\(949\) −7.31202e9 −0.277719
\(950\) 3.41553e10 1.29249
\(951\) 1.74284e10 0.657091
\(952\) 2.63539e10 0.989954
\(953\) 2.33081e10 0.872333 0.436166 0.899866i \(-0.356336\pi\)
0.436166 + 0.899866i \(0.356336\pi\)
\(954\) −2.57745e9 −0.0961107
\(955\) 4.38016e10 1.62734
\(956\) −4.39451e9 −0.162670
\(957\) −2.85083e10 −1.05143
\(958\) 3.10598e9 0.114135
\(959\) 5.64726e9 0.206763
\(960\) −2.50028e10 −0.912094
\(961\) 2.21597e10 0.805438
\(962\) 1.71796e10 0.622159
\(963\) 1.44474e10 0.521311
\(964\) 7.39225e9 0.265771
\(965\) −3.37296e10 −1.20827
\(966\) 2.05427e9 0.0733227
\(967\) 3.20553e9 0.114001 0.0570003 0.998374i \(-0.481846\pi\)
0.0570003 + 0.998374i \(0.481846\pi\)
\(968\) −1.65342e10 −0.585893
\(969\) 3.79771e10 1.34088
\(970\) −6.11647e10 −2.15179
\(971\) −2.31196e10 −0.810424 −0.405212 0.914223i \(-0.632802\pi\)
−0.405212 + 0.914223i \(0.632802\pi\)
\(972\) −3.41457e8 −0.0119263
\(973\) −2.13845e10 −0.744224
\(974\) −8.80265e9 −0.305251
\(975\) 1.13702e10 0.392873
\(976\) 3.56090e10 1.22599
\(977\) 9.17629e9 0.314801 0.157400 0.987535i \(-0.449689\pi\)
0.157400 + 0.987535i \(0.449689\pi\)
\(978\) 8.34119e9 0.285129
\(979\) −3.03219e10 −1.03280
\(980\) −5.37462e9 −0.182413
\(981\) 7.14684e9 0.241698
\(982\) −1.11868e10 −0.376977
\(983\) −2.25628e10 −0.757626 −0.378813 0.925473i \(-0.623668\pi\)
−0.378813 + 0.925473i \(0.623668\pi\)
\(984\) 8.78598e9 0.293973
\(985\) −3.36066e10 −1.12046
\(986\) 6.60245e10 2.19349
\(987\) −1.69569e10 −0.561353
\(988\) −5.23615e9 −0.172728
\(989\) 2.91675e9 0.0958764
\(990\) 1.62517e10 0.532322
\(991\) 4.42802e10 1.44528 0.722640 0.691224i \(-0.242928\pi\)
0.722640 + 0.691224i \(0.242928\pi\)
\(992\) −1.51481e10 −0.492682
\(993\) −2.78778e10 −0.903515
\(994\) −7.73707e9 −0.249876
\(995\) 4.56851e10 1.47026
\(996\) −1.30640e9 −0.0418957
\(997\) 2.83830e10 0.907036 0.453518 0.891247i \(-0.350169\pi\)
0.453518 + 0.891247i \(0.350169\pi\)
\(998\) 2.73012e10 0.869409
\(999\) 6.29468e9 0.199754
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.d.1.13 18
3.2 odd 2 531.8.a.e.1.6 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.13 18 1.1 even 1 trivial
531.8.a.e.1.6 18 3.2 odd 2