Properties

Label 177.8.a.d.1.14
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 1.14
Root \(13.6971\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+14.6971 q^{2} +27.0000 q^{3} +88.0043 q^{4} +457.882 q^{5} +396.821 q^{6} +1576.42 q^{7} -587.821 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+14.6971 q^{2} +27.0000 q^{3} +88.0043 q^{4} +457.882 q^{5} +396.821 q^{6} +1576.42 q^{7} -587.821 q^{8} +729.000 q^{9} +6729.53 q^{10} +6841.63 q^{11} +2376.11 q^{12} +4116.85 q^{13} +23168.8 q^{14} +12362.8 q^{15} -19903.8 q^{16} -34857.5 q^{17} +10714.2 q^{18} -30826.5 q^{19} +40295.5 q^{20} +42563.4 q^{21} +100552. q^{22} -51444.0 q^{23} -15871.2 q^{24} +131531. q^{25} +60505.7 q^{26} +19683.0 q^{27} +138732. q^{28} +106767. q^{29} +181697. q^{30} -236039. q^{31} -217287. q^{32} +184724. q^{33} -512304. q^{34} +721815. q^{35} +64155.1 q^{36} -320825. q^{37} -453059. q^{38} +111155. q^{39} -269152. q^{40} -227909. q^{41} +625558. q^{42} +156061. q^{43} +602092. q^{44} +333796. q^{45} -756076. q^{46} +902640. q^{47} -537402. q^{48} +1.66156e6 q^{49} +1.93312e6 q^{50} -941152. q^{51} +362300. q^{52} -1.28791e6 q^{53} +289283. q^{54} +3.13266e6 q^{55} -926654. q^{56} -832315. q^{57} +1.56916e6 q^{58} +205379. q^{59} +1.08798e6 q^{60} +719101. q^{61} -3.46908e6 q^{62} +1.14921e6 q^{63} -645794. q^{64} +1.88503e6 q^{65} +2.71490e6 q^{66} -2.73492e6 q^{67} -3.06761e6 q^{68} -1.38899e6 q^{69} +1.06086e7 q^{70} +2.81839e6 q^{71} -428521. q^{72} -880677. q^{73} -4.71519e6 q^{74} +3.55133e6 q^{75} -2.71286e6 q^{76} +1.07853e7 q^{77} +1.63365e6 q^{78} +2.91979e6 q^{79} -9.11359e6 q^{80} +531441. q^{81} -3.34960e6 q^{82} -6.61990e6 q^{83} +3.74576e6 q^{84} -1.59606e7 q^{85} +2.29364e6 q^{86} +2.88271e6 q^{87} -4.02165e6 q^{88} -6.36510e6 q^{89} +4.90583e6 q^{90} +6.48989e6 q^{91} -4.52729e6 q^{92} -6.37305e6 q^{93} +1.32662e7 q^{94} -1.41149e7 q^{95} -5.86674e6 q^{96} +1.73556e7 q^{97} +2.44201e7 q^{98} +4.98755e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.6971 1.29905 0.649525 0.760340i \(-0.274968\pi\)
0.649525 + 0.760340i \(0.274968\pi\)
\(3\) 27.0000 0.577350
\(4\) 88.0043 0.687533
\(5\) 457.882 1.63817 0.819084 0.573674i \(-0.194482\pi\)
0.819084 + 0.573674i \(0.194482\pi\)
\(6\) 396.821 0.750007
\(7\) 1576.42 1.73712 0.868559 0.495585i \(-0.165046\pi\)
0.868559 + 0.495585i \(0.165046\pi\)
\(8\) −587.821 −0.405910
\(9\) 729.000 0.333333
\(10\) 6729.53 2.12806
\(11\) 6841.63 1.54983 0.774917 0.632064i \(-0.217792\pi\)
0.774917 + 0.632064i \(0.217792\pi\)
\(12\) 2376.11 0.396947
\(13\) 4116.85 0.519713 0.259856 0.965647i \(-0.416325\pi\)
0.259856 + 0.965647i \(0.416325\pi\)
\(14\) 23168.8 2.25661
\(15\) 12362.8 0.945797
\(16\) −19903.8 −1.21483
\(17\) −34857.5 −1.72078 −0.860389 0.509637i \(-0.829780\pi\)
−0.860389 + 0.509637i \(0.829780\pi\)
\(18\) 10714.2 0.433017
\(19\) −30826.5 −1.03107 −0.515533 0.856870i \(-0.672406\pi\)
−0.515533 + 0.856870i \(0.672406\pi\)
\(20\) 40295.5 1.12629
\(21\) 42563.4 1.00293
\(22\) 100552. 2.01331
\(23\) −51444.0 −0.881632 −0.440816 0.897598i \(-0.645311\pi\)
−0.440816 + 0.897598i \(0.645311\pi\)
\(24\) −15871.2 −0.234352
\(25\) 131531. 1.68359
\(26\) 60505.7 0.675133
\(27\) 19683.0 0.192450
\(28\) 138732. 1.19433
\(29\) 106767. 0.812913 0.406457 0.913670i \(-0.366764\pi\)
0.406457 + 0.913670i \(0.366764\pi\)
\(30\) 181697. 1.22864
\(31\) −236039. −1.42304 −0.711521 0.702665i \(-0.751993\pi\)
−0.711521 + 0.702665i \(0.751993\pi\)
\(32\) −217287. −1.17222
\(33\) 184724. 0.894797
\(34\) −512304. −2.23538
\(35\) 721815. 2.84569
\(36\) 64155.1 0.229178
\(37\) −320825. −1.04127 −0.520633 0.853781i \(-0.674304\pi\)
−0.520633 + 0.853781i \(0.674304\pi\)
\(38\) −453059. −1.33941
\(39\) 111155. 0.300056
\(40\) −269152. −0.664949
\(41\) −227909. −0.516439 −0.258219 0.966086i \(-0.583136\pi\)
−0.258219 + 0.966086i \(0.583136\pi\)
\(42\) 625558. 1.30285
\(43\) 156061. 0.299333 0.149667 0.988737i \(-0.452180\pi\)
0.149667 + 0.988737i \(0.452180\pi\)
\(44\) 602092. 1.06556
\(45\) 333796. 0.546056
\(46\) −756076. −1.14528
\(47\) 902640. 1.26815 0.634077 0.773270i \(-0.281380\pi\)
0.634077 + 0.773270i \(0.281380\pi\)
\(48\) −537402. −0.701383
\(49\) 1.66156e6 2.01758
\(50\) 1.93312e6 2.18707
\(51\) −941152. −0.993492
\(52\) 362300. 0.357320
\(53\) −1.28791e6 −1.18829 −0.594143 0.804360i \(-0.702509\pi\)
−0.594143 + 0.804360i \(0.702509\pi\)
\(54\) 289283. 0.250002
\(55\) 3.13266e6 2.53889
\(56\) −926654. −0.705114
\(57\) −832315. −0.595286
\(58\) 1.56916e6 1.05602
\(59\) 205379. 0.130189
\(60\) 1.08798e6 0.650267
\(61\) 719101. 0.405635 0.202818 0.979217i \(-0.434990\pi\)
0.202818 + 0.979217i \(0.434990\pi\)
\(62\) −3.46908e6 −1.84860
\(63\) 1.14921e6 0.579039
\(64\) −645794. −0.307939
\(65\) 1.88503e6 0.851377
\(66\) 2.71490e6 1.16239
\(67\) −2.73492e6 −1.11092 −0.555460 0.831544i \(-0.687458\pi\)
−0.555460 + 0.831544i \(0.687458\pi\)
\(68\) −3.06761e6 −1.18309
\(69\) −1.38899e6 −0.509010
\(70\) 1.06086e7 3.69670
\(71\) 2.81839e6 0.934538 0.467269 0.884115i \(-0.345238\pi\)
0.467269 + 0.884115i \(0.345238\pi\)
\(72\) −428521. −0.135303
\(73\) −880677. −0.264964 −0.132482 0.991185i \(-0.542295\pi\)
−0.132482 + 0.991185i \(0.542295\pi\)
\(74\) −4.71519e6 −1.35266
\(75\) 3.55133e6 0.972023
\(76\) −2.71286e6 −0.708892
\(77\) 1.07853e7 2.69224
\(78\) 1.63365e6 0.389788
\(79\) 2.91979e6 0.666281 0.333140 0.942877i \(-0.391892\pi\)
0.333140 + 0.942877i \(0.391892\pi\)
\(80\) −9.11359e6 −1.99010
\(81\) 531441. 0.111111
\(82\) −3.34960e6 −0.670880
\(83\) −6.61990e6 −1.27080 −0.635402 0.772182i \(-0.719165\pi\)
−0.635402 + 0.772182i \(0.719165\pi\)
\(84\) 3.74576e6 0.689545
\(85\) −1.59606e7 −2.81892
\(86\) 2.29364e6 0.388849
\(87\) 2.88271e6 0.469336
\(88\) −4.02165e6 −0.629093
\(89\) −6.36510e6 −0.957062 −0.478531 0.878071i \(-0.658831\pi\)
−0.478531 + 0.878071i \(0.658831\pi\)
\(90\) 4.90583e6 0.709354
\(91\) 6.48989e6 0.902803
\(92\) −4.52729e6 −0.606151
\(93\) −6.37305e6 −0.821593
\(94\) 1.32662e7 1.64740
\(95\) −1.41149e7 −1.68906
\(96\) −5.86674e6 −0.676780
\(97\) 1.73556e7 1.93081 0.965403 0.260764i \(-0.0839744\pi\)
0.965403 + 0.260764i \(0.0839744\pi\)
\(98\) 2.44201e7 2.62094
\(99\) 4.98755e6 0.516611
\(100\) 1.15753e7 1.15753
\(101\) 1.48075e7 1.43007 0.715036 0.699087i \(-0.246410\pi\)
0.715036 + 0.699087i \(0.246410\pi\)
\(102\) −1.38322e7 −1.29060
\(103\) 8.68688e6 0.783310 0.391655 0.920112i \(-0.371903\pi\)
0.391655 + 0.920112i \(0.371903\pi\)
\(104\) −2.41997e6 −0.210957
\(105\) 1.94890e7 1.64296
\(106\) −1.89286e7 −1.54364
\(107\) −1.01246e7 −0.798974 −0.399487 0.916739i \(-0.630812\pi\)
−0.399487 + 0.916739i \(0.630812\pi\)
\(108\) 1.73219e6 0.132316
\(109\) −4.44690e6 −0.328900 −0.164450 0.986385i \(-0.552585\pi\)
−0.164450 + 0.986385i \(0.552585\pi\)
\(110\) 4.60409e7 3.29814
\(111\) −8.66227e6 −0.601175
\(112\) −3.13768e7 −2.11031
\(113\) −2.11640e6 −0.137982 −0.0689910 0.997617i \(-0.521978\pi\)
−0.0689910 + 0.997617i \(0.521978\pi\)
\(114\) −1.22326e7 −0.773307
\(115\) −2.35553e7 −1.44426
\(116\) 9.39595e6 0.558905
\(117\) 3.00118e6 0.173238
\(118\) 3.01847e6 0.169122
\(119\) −5.49501e7 −2.98920
\(120\) −7.26712e6 −0.383909
\(121\) 2.73207e7 1.40198
\(122\) 1.05687e7 0.526941
\(123\) −6.15356e6 −0.298166
\(124\) −2.07724e7 −0.978388
\(125\) 2.44535e7 1.11984
\(126\) 1.68901e7 0.752202
\(127\) 1.43195e7 0.620317 0.310159 0.950685i \(-0.399618\pi\)
0.310159 + 0.950685i \(0.399618\pi\)
\(128\) 1.83214e7 0.772189
\(129\) 4.21365e6 0.172820
\(130\) 2.77045e7 1.10598
\(131\) −4.80639e7 −1.86797 −0.933985 0.357312i \(-0.883693\pi\)
−0.933985 + 0.357312i \(0.883693\pi\)
\(132\) 1.62565e7 0.615202
\(133\) −4.85956e7 −1.79108
\(134\) −4.01953e7 −1.44314
\(135\) 9.01249e6 0.315266
\(136\) 2.04900e7 0.698482
\(137\) −1.81664e7 −0.603598 −0.301799 0.953372i \(-0.597587\pi\)
−0.301799 + 0.953372i \(0.597587\pi\)
\(138\) −2.04141e7 −0.661230
\(139\) 2.63273e7 0.831484 0.415742 0.909483i \(-0.363522\pi\)
0.415742 + 0.909483i \(0.363522\pi\)
\(140\) 6.35228e7 1.95651
\(141\) 2.43713e7 0.732170
\(142\) 4.14221e7 1.21401
\(143\) 2.81659e7 0.805468
\(144\) −1.45099e7 −0.404944
\(145\) 4.88867e7 1.33169
\(146\) −1.29434e7 −0.344201
\(147\) 4.48622e7 1.16485
\(148\) −2.82339e7 −0.715905
\(149\) 6.64472e7 1.64560 0.822801 0.568329i \(-0.192410\pi\)
0.822801 + 0.568329i \(0.192410\pi\)
\(150\) 5.21942e7 1.26271
\(151\) −6.98756e7 −1.65160 −0.825802 0.563960i \(-0.809277\pi\)
−0.825802 + 0.563960i \(0.809277\pi\)
\(152\) 1.81205e7 0.418520
\(153\) −2.54111e7 −0.573593
\(154\) 1.58512e8 3.49736
\(155\) −1.08078e8 −2.33118
\(156\) 9.78211e6 0.206299
\(157\) −2.65494e7 −0.547528 −0.273764 0.961797i \(-0.588269\pi\)
−0.273764 + 0.961797i \(0.588269\pi\)
\(158\) 4.29124e7 0.865533
\(159\) −3.47736e7 −0.686057
\(160\) −9.94916e7 −1.92029
\(161\) −8.10974e7 −1.53150
\(162\) 7.81063e6 0.144339
\(163\) 5.75891e7 1.04156 0.520779 0.853691i \(-0.325642\pi\)
0.520779 + 0.853691i \(0.325642\pi\)
\(164\) −2.00570e7 −0.355069
\(165\) 8.45817e7 1.46583
\(166\) −9.72933e7 −1.65084
\(167\) 4.98500e6 0.0828242 0.0414121 0.999142i \(-0.486814\pi\)
0.0414121 + 0.999142i \(0.486814\pi\)
\(168\) −2.50197e7 −0.407098
\(169\) −4.58001e7 −0.729899
\(170\) −2.34574e8 −3.66193
\(171\) −2.24725e7 −0.343689
\(172\) 1.37340e7 0.205801
\(173\) 8.44843e7 1.24055 0.620276 0.784384i \(-0.287021\pi\)
0.620276 + 0.784384i \(0.287021\pi\)
\(174\) 4.23674e7 0.609691
\(175\) 2.07348e8 2.92460
\(176\) −1.36174e8 −1.88279
\(177\) 5.54523e6 0.0751646
\(178\) −9.35484e7 −1.24327
\(179\) −7.90602e6 −0.103032 −0.0515160 0.998672i \(-0.516405\pi\)
−0.0515160 + 0.998672i \(0.516405\pi\)
\(180\) 2.93755e7 0.375432
\(181\) 4.23307e7 0.530616 0.265308 0.964164i \(-0.414526\pi\)
0.265308 + 0.964164i \(0.414526\pi\)
\(182\) 9.53825e7 1.17279
\(183\) 1.94157e7 0.234194
\(184\) 3.02398e7 0.357863
\(185\) −1.46900e8 −1.70577
\(186\) −9.36652e7 −1.06729
\(187\) −2.38482e8 −2.66692
\(188\) 7.94362e7 0.871899
\(189\) 3.10287e7 0.334309
\(190\) −2.07448e8 −2.19417
\(191\) −8.15774e7 −0.847136 −0.423568 0.905864i \(-0.639223\pi\)
−0.423568 + 0.905864i \(0.639223\pi\)
\(192\) −1.74365e7 −0.177789
\(193\) 3.34817e7 0.335240 0.167620 0.985852i \(-0.446392\pi\)
0.167620 + 0.985852i \(0.446392\pi\)
\(194\) 2.55077e8 2.50821
\(195\) 5.08958e7 0.491543
\(196\) 1.46225e8 1.38715
\(197\) −5.69368e7 −0.530593 −0.265296 0.964167i \(-0.585470\pi\)
−0.265296 + 0.964167i \(0.585470\pi\)
\(198\) 7.33024e7 0.671104
\(199\) 3.68438e7 0.331420 0.165710 0.986175i \(-0.447008\pi\)
0.165710 + 0.986175i \(0.447008\pi\)
\(200\) −7.73165e7 −0.683388
\(201\) −7.38428e7 −0.641389
\(202\) 2.17628e8 1.85774
\(203\) 1.68310e8 1.41213
\(204\) −8.28254e7 −0.683059
\(205\) −1.04356e8 −0.846013
\(206\) 1.27672e8 1.01756
\(207\) −3.75027e7 −0.293877
\(208\) −8.19410e7 −0.631363
\(209\) −2.10903e8 −1.59798
\(210\) 2.86432e8 2.13429
\(211\) 6.42783e6 0.0471059 0.0235530 0.999723i \(-0.492502\pi\)
0.0235530 + 0.999723i \(0.492502\pi\)
\(212\) −1.13342e8 −0.816986
\(213\) 7.60965e7 0.539556
\(214\) −1.48801e8 −1.03791
\(215\) 7.14575e7 0.490358
\(216\) −1.15701e7 −0.0781175
\(217\) −3.72097e8 −2.47199
\(218\) −6.53564e7 −0.427258
\(219\) −2.37783e7 −0.152977
\(220\) 2.75687e8 1.74557
\(221\) −1.43503e8 −0.894311
\(222\) −1.27310e8 −0.780957
\(223\) −5.93351e7 −0.358298 −0.179149 0.983822i \(-0.557334\pi\)
−0.179149 + 0.983822i \(0.557334\pi\)
\(224\) −3.42536e8 −2.03628
\(225\) 9.58859e7 0.561198
\(226\) −3.11049e7 −0.179246
\(227\) 2.49947e8 1.41827 0.709134 0.705074i \(-0.249086\pi\)
0.709134 + 0.705074i \(0.249086\pi\)
\(228\) −7.32473e7 −0.409279
\(229\) −1.83888e8 −1.01188 −0.505941 0.862568i \(-0.668855\pi\)
−0.505941 + 0.862568i \(0.668855\pi\)
\(230\) −3.46194e8 −1.87617
\(231\) 2.91203e8 1.55437
\(232\) −6.27599e7 −0.329970
\(233\) −1.34216e8 −0.695120 −0.347560 0.937658i \(-0.612990\pi\)
−0.347560 + 0.937658i \(0.612990\pi\)
\(234\) 4.41087e7 0.225044
\(235\) 4.13303e8 2.07745
\(236\) 1.80742e7 0.0895092
\(237\) 7.88344e7 0.384677
\(238\) −8.07607e8 −3.88312
\(239\) −3.79885e8 −1.79994 −0.899972 0.435947i \(-0.856413\pi\)
−0.899972 + 0.435947i \(0.856413\pi\)
\(240\) −2.46067e8 −1.14898
\(241\) −2.48361e7 −0.114294 −0.0571470 0.998366i \(-0.518200\pi\)
−0.0571470 + 0.998366i \(0.518200\pi\)
\(242\) 4.01534e8 1.82125
\(243\) 1.43489e7 0.0641500
\(244\) 6.32840e7 0.278888
\(245\) 7.60800e8 3.30514
\(246\) −9.04393e7 −0.387333
\(247\) −1.26908e8 −0.535858
\(248\) 1.38749e8 0.577627
\(249\) −1.78737e8 −0.733699
\(250\) 3.59395e8 1.45473
\(251\) −2.30663e8 −0.920705 −0.460352 0.887736i \(-0.652277\pi\)
−0.460352 + 0.887736i \(0.652277\pi\)
\(252\) 1.01136e8 0.398109
\(253\) −3.51960e8 −1.36638
\(254\) 2.10454e8 0.805824
\(255\) −4.30937e8 −1.62751
\(256\) 3.51933e8 1.31105
\(257\) 2.98007e8 1.09512 0.547558 0.836768i \(-0.315558\pi\)
0.547558 + 0.836768i \(0.315558\pi\)
\(258\) 6.19283e7 0.224502
\(259\) −5.05755e8 −1.80880
\(260\) 1.65891e8 0.585350
\(261\) 7.78332e7 0.270971
\(262\) −7.06400e8 −2.42659
\(263\) 2.77082e8 0.939211 0.469605 0.882876i \(-0.344396\pi\)
0.469605 + 0.882876i \(0.344396\pi\)
\(264\) −1.08585e8 −0.363207
\(265\) −5.89712e8 −1.94661
\(266\) −7.14213e8 −2.32671
\(267\) −1.71858e8 −0.552560
\(268\) −2.40684e8 −0.763794
\(269\) 2.71094e8 0.849155 0.424577 0.905392i \(-0.360423\pi\)
0.424577 + 0.905392i \(0.360423\pi\)
\(270\) 1.32457e8 0.409546
\(271\) −4.89470e8 −1.49394 −0.746970 0.664858i \(-0.768492\pi\)
−0.746970 + 0.664858i \(0.768492\pi\)
\(272\) 6.93797e8 2.09046
\(273\) 1.75227e8 0.521233
\(274\) −2.66994e8 −0.784104
\(275\) 8.99884e8 2.60929
\(276\) −1.22237e8 −0.349961
\(277\) −2.76753e8 −0.782371 −0.391186 0.920312i \(-0.627935\pi\)
−0.391186 + 0.920312i \(0.627935\pi\)
\(278\) 3.86934e8 1.08014
\(279\) −1.72072e8 −0.474347
\(280\) −4.24298e8 −1.15510
\(281\) 3.28075e8 0.882065 0.441032 0.897491i \(-0.354612\pi\)
0.441032 + 0.897491i \(0.354612\pi\)
\(282\) 3.58187e8 0.951126
\(283\) 2.47479e8 0.649061 0.324531 0.945875i \(-0.394794\pi\)
0.324531 + 0.945875i \(0.394794\pi\)
\(284\) 2.48030e8 0.642526
\(285\) −3.81102e8 −0.975179
\(286\) 4.13957e8 1.04634
\(287\) −3.59282e8 −0.897115
\(288\) −1.58402e8 −0.390739
\(289\) 8.04706e8 1.96108
\(290\) 7.18492e8 1.72993
\(291\) 4.68601e8 1.11475
\(292\) −7.75033e7 −0.182171
\(293\) 3.73118e8 0.866582 0.433291 0.901254i \(-0.357352\pi\)
0.433291 + 0.901254i \(0.357352\pi\)
\(294\) 6.59344e8 1.51320
\(295\) 9.40393e7 0.213271
\(296\) 1.88587e8 0.422661
\(297\) 1.34664e8 0.298266
\(298\) 9.76581e8 2.13772
\(299\) −2.11787e8 −0.458195
\(300\) 3.12532e8 0.668298
\(301\) 2.46018e8 0.519977
\(302\) −1.02697e9 −2.14552
\(303\) 3.99804e8 0.825653
\(304\) 6.13564e8 1.25257
\(305\) 3.29263e8 0.664498
\(306\) −3.73469e8 −0.745126
\(307\) −5.11504e8 −1.00894 −0.504469 0.863430i \(-0.668312\pi\)
−0.504469 + 0.863430i \(0.668312\pi\)
\(308\) 9.49151e8 1.85101
\(309\) 2.34546e8 0.452244
\(310\) −1.58843e9 −3.02832
\(311\) 6.07087e6 0.0114443 0.00572215 0.999984i \(-0.498179\pi\)
0.00572215 + 0.999984i \(0.498179\pi\)
\(312\) −6.53392e7 −0.121796
\(313\) 9.00976e6 0.0166076 0.00830382 0.999966i \(-0.497357\pi\)
0.00830382 + 0.999966i \(0.497357\pi\)
\(314\) −3.90199e8 −0.711266
\(315\) 5.26203e8 0.948564
\(316\) 2.56954e8 0.458090
\(317\) 1.54129e8 0.271755 0.135878 0.990726i \(-0.456615\pi\)
0.135878 + 0.990726i \(0.456615\pi\)
\(318\) −5.11071e8 −0.891223
\(319\) 7.30460e8 1.25988
\(320\) −2.95698e8 −0.504455
\(321\) −2.73363e8 −0.461288
\(322\) −1.19190e9 −1.98949
\(323\) 1.07453e9 1.77424
\(324\) 4.67691e7 0.0763926
\(325\) 5.41492e8 0.874985
\(326\) 8.46391e8 1.35304
\(327\) −1.20066e8 −0.189891
\(328\) 1.33970e8 0.209628
\(329\) 1.42294e9 2.20294
\(330\) 1.24310e9 1.90418
\(331\) 6.66857e8 1.01073 0.505365 0.862906i \(-0.331358\pi\)
0.505365 + 0.862906i \(0.331358\pi\)
\(332\) −5.82580e8 −0.873719
\(333\) −2.33881e8 −0.347089
\(334\) 7.32649e7 0.107593
\(335\) −1.25227e9 −1.81987
\(336\) −8.47173e8 −1.21839
\(337\) 7.12103e8 1.01353 0.506767 0.862083i \(-0.330840\pi\)
0.506767 + 0.862083i \(0.330840\pi\)
\(338\) −6.73127e8 −0.948176
\(339\) −5.71427e7 −0.0796640
\(340\) −1.40460e9 −1.93810
\(341\) −1.61489e9 −2.20548
\(342\) −3.30280e8 −0.446469
\(343\) 1.32108e9 1.76766
\(344\) −9.17359e7 −0.121502
\(345\) −6.35992e8 −0.833844
\(346\) 1.24167e9 1.61154
\(347\) 6.85187e8 0.880351 0.440175 0.897912i \(-0.354916\pi\)
0.440175 + 0.897912i \(0.354916\pi\)
\(348\) 2.53691e8 0.322684
\(349\) −4.10968e8 −0.517511 −0.258755 0.965943i \(-0.583312\pi\)
−0.258755 + 0.965943i \(0.583312\pi\)
\(350\) 3.04741e9 3.79921
\(351\) 8.10320e7 0.100019
\(352\) −1.48659e9 −1.81674
\(353\) 7.36899e8 0.891654 0.445827 0.895119i \(-0.352910\pi\)
0.445827 + 0.895119i \(0.352910\pi\)
\(354\) 8.14988e7 0.0976426
\(355\) 1.29049e9 1.53093
\(356\) −5.60156e8 −0.658012
\(357\) −1.48365e9 −1.72581
\(358\) −1.16195e8 −0.133844
\(359\) 1.53809e9 1.75449 0.877246 0.480041i \(-0.159378\pi\)
0.877246 + 0.480041i \(0.159378\pi\)
\(360\) −1.96212e8 −0.221650
\(361\) 5.64007e7 0.0630971
\(362\) 6.22138e8 0.689298
\(363\) 7.37658e8 0.809435
\(364\) 5.71138e8 0.620707
\(365\) −4.03246e8 −0.434055
\(366\) 2.85355e8 0.304229
\(367\) −9.99795e8 −1.05580 −0.527898 0.849308i \(-0.677020\pi\)
−0.527898 + 0.849308i \(0.677020\pi\)
\(368\) 1.02393e9 1.07103
\(369\) −1.66146e8 −0.172146
\(370\) −2.15900e9 −2.21588
\(371\) −2.03029e9 −2.06419
\(372\) −5.60855e8 −0.564873
\(373\) −9.19979e8 −0.917903 −0.458952 0.888461i \(-0.651775\pi\)
−0.458952 + 0.888461i \(0.651775\pi\)
\(374\) −3.50499e9 −3.46446
\(375\) 6.60245e8 0.646540
\(376\) −5.30591e8 −0.514757
\(377\) 4.39544e8 0.422481
\(378\) 4.56032e8 0.434284
\(379\) 1.21564e9 1.14701 0.573505 0.819202i \(-0.305583\pi\)
0.573505 + 0.819202i \(0.305583\pi\)
\(380\) −1.24217e9 −1.16128
\(381\) 3.86626e8 0.358140
\(382\) −1.19895e9 −1.10047
\(383\) 3.85577e8 0.350683 0.175342 0.984508i \(-0.443897\pi\)
0.175342 + 0.984508i \(0.443897\pi\)
\(384\) 4.94678e8 0.445824
\(385\) 4.93839e9 4.41035
\(386\) 4.92083e8 0.435494
\(387\) 1.13768e8 0.0997777
\(388\) 1.52737e9 1.32749
\(389\) −1.39089e9 −1.19804 −0.599019 0.800735i \(-0.704443\pi\)
−0.599019 + 0.800735i \(0.704443\pi\)
\(390\) 7.48020e8 0.638539
\(391\) 1.79321e9 1.51709
\(392\) −9.76702e8 −0.818957
\(393\) −1.29773e9 −1.07847
\(394\) −8.36805e8 −0.689267
\(395\) 1.33692e9 1.09148
\(396\) 4.38925e8 0.355187
\(397\) −1.34669e8 −0.108019 −0.0540096 0.998540i \(-0.517200\pi\)
−0.0540096 + 0.998540i \(0.517200\pi\)
\(398\) 5.41497e8 0.430532
\(399\) −1.31208e9 −1.03408
\(400\) −2.61796e9 −2.04528
\(401\) 7.92234e8 0.613547 0.306774 0.951782i \(-0.400751\pi\)
0.306774 + 0.951782i \(0.400751\pi\)
\(402\) −1.08527e9 −0.833197
\(403\) −9.71737e8 −0.739573
\(404\) 1.30313e9 0.983223
\(405\) 2.43337e8 0.182019
\(406\) 2.47367e9 1.83442
\(407\) −2.19496e9 −1.61379
\(408\) 5.53229e8 0.403269
\(409\) −1.98450e9 −1.43424 −0.717118 0.696952i \(-0.754539\pi\)
−0.717118 + 0.696952i \(0.754539\pi\)
\(410\) −1.53372e9 −1.09901
\(411\) −4.90494e8 −0.348487
\(412\) 7.64482e8 0.538551
\(413\) 3.23764e8 0.226154
\(414\) −5.51180e8 −0.381761
\(415\) −3.03113e9 −2.08179
\(416\) −8.94537e8 −0.609216
\(417\) 7.10836e8 0.480058
\(418\) −3.09966e9 −2.07586
\(419\) 1.12010e9 0.743886 0.371943 0.928256i \(-0.378692\pi\)
0.371943 + 0.928256i \(0.378692\pi\)
\(420\) 1.71512e9 1.12959
\(421\) 1.60154e9 1.04605 0.523024 0.852318i \(-0.324804\pi\)
0.523024 + 0.852318i \(0.324804\pi\)
\(422\) 9.44703e7 0.0611930
\(423\) 6.58025e8 0.422718
\(424\) 7.57062e8 0.482337
\(425\) −4.58483e9 −2.89709
\(426\) 1.11840e9 0.700910
\(427\) 1.13361e9 0.704636
\(428\) −8.91004e8 −0.549321
\(429\) 7.60481e8 0.465037
\(430\) 1.05022e9 0.637000
\(431\) −1.42222e9 −0.855648 −0.427824 0.903862i \(-0.640720\pi\)
−0.427824 + 0.903862i \(0.640720\pi\)
\(432\) −3.91766e8 −0.233794
\(433\) 1.13304e9 0.670713 0.335356 0.942091i \(-0.391143\pi\)
0.335356 + 0.942091i \(0.391143\pi\)
\(434\) −5.46874e9 −3.21124
\(435\) 1.31994e9 0.768851
\(436\) −3.91346e8 −0.226130
\(437\) 1.58584e9 0.909020
\(438\) −3.49471e8 −0.198725
\(439\) −1.92938e9 −1.08841 −0.544204 0.838953i \(-0.683168\pi\)
−0.544204 + 0.838953i \(0.683168\pi\)
\(440\) −1.84144e9 −1.03056
\(441\) 1.21128e9 0.672527
\(442\) −2.10908e9 −1.16175
\(443\) −8.33711e7 −0.0455620 −0.0227810 0.999740i \(-0.507252\pi\)
−0.0227810 + 0.999740i \(0.507252\pi\)
\(444\) −7.62316e8 −0.413328
\(445\) −2.91446e9 −1.56783
\(446\) −8.72053e8 −0.465448
\(447\) 1.79408e9 0.950089
\(448\) −1.01804e9 −0.534926
\(449\) 2.88898e9 1.50620 0.753099 0.657907i \(-0.228558\pi\)
0.753099 + 0.657907i \(0.228558\pi\)
\(450\) 1.40924e9 0.729025
\(451\) −1.55927e9 −0.800394
\(452\) −1.86252e8 −0.0948673
\(453\) −1.88664e9 −0.953554
\(454\) 3.67350e9 1.84240
\(455\) 2.97160e9 1.47894
\(456\) 4.89252e8 0.241633
\(457\) −2.23596e9 −1.09587 −0.547933 0.836522i \(-0.684585\pi\)
−0.547933 + 0.836522i \(0.684585\pi\)
\(458\) −2.70262e9 −1.31449
\(459\) −6.86100e8 −0.331164
\(460\) −2.07296e9 −0.992977
\(461\) −1.26440e9 −0.601077 −0.300539 0.953770i \(-0.597166\pi\)
−0.300539 + 0.953770i \(0.597166\pi\)
\(462\) 4.27983e9 2.01920
\(463\) 2.39212e9 1.12008 0.560041 0.828465i \(-0.310785\pi\)
0.560041 + 0.828465i \(0.310785\pi\)
\(464\) −2.12507e9 −0.987553
\(465\) −2.91810e9 −1.34591
\(466\) −1.97259e9 −0.902996
\(467\) 5.83299e8 0.265022 0.132511 0.991182i \(-0.457696\pi\)
0.132511 + 0.991182i \(0.457696\pi\)
\(468\) 2.64117e8 0.119107
\(469\) −4.31138e9 −1.92980
\(470\) 6.07434e9 2.69871
\(471\) −7.16834e8 −0.316115
\(472\) −1.20726e8 −0.0528450
\(473\) 1.06771e9 0.463916
\(474\) 1.15864e9 0.499715
\(475\) −4.05463e9 −1.73590
\(476\) −4.83585e9 −2.05517
\(477\) −9.38888e8 −0.396095
\(478\) −5.58320e9 −2.33822
\(479\) 3.79366e9 1.57719 0.788595 0.614913i \(-0.210809\pi\)
0.788595 + 0.614913i \(0.210809\pi\)
\(480\) −2.68627e9 −1.10868
\(481\) −1.32079e9 −0.541159
\(482\) −3.65018e8 −0.148474
\(483\) −2.18963e9 −0.884211
\(484\) 2.40434e9 0.963909
\(485\) 7.94681e9 3.16298
\(486\) 2.10887e8 0.0833342
\(487\) −2.39967e9 −0.941455 −0.470728 0.882279i \(-0.656009\pi\)
−0.470728 + 0.882279i \(0.656009\pi\)
\(488\) −4.22703e8 −0.164651
\(489\) 1.55491e9 0.601344
\(490\) 1.11815e10 4.29354
\(491\) 4.01253e9 1.52979 0.764897 0.644152i \(-0.222790\pi\)
0.764897 + 0.644152i \(0.222790\pi\)
\(492\) −5.41539e8 −0.204999
\(493\) −3.72163e9 −1.39884
\(494\) −1.86518e9 −0.696107
\(495\) 2.28371e9 0.846295
\(496\) 4.69807e9 1.72876
\(497\) 4.44297e9 1.62340
\(498\) −2.62692e9 −0.953112
\(499\) 2.58807e9 0.932445 0.466223 0.884667i \(-0.345615\pi\)
0.466223 + 0.884667i \(0.345615\pi\)
\(500\) 2.15201e9 0.769928
\(501\) 1.34595e8 0.0478186
\(502\) −3.39008e9 −1.19604
\(503\) −5.50700e9 −1.92942 −0.964712 0.263309i \(-0.915186\pi\)
−0.964712 + 0.263309i \(0.915186\pi\)
\(504\) −6.75531e8 −0.235038
\(505\) 6.78010e9 2.34270
\(506\) −5.17279e9 −1.77500
\(507\) −1.23660e9 −0.421407
\(508\) 1.26017e9 0.426489
\(509\) 1.30017e9 0.437007 0.218503 0.975836i \(-0.429883\pi\)
0.218503 + 0.975836i \(0.429883\pi\)
\(510\) −6.33351e9 −2.11421
\(511\) −1.38832e9 −0.460273
\(512\) 2.82725e9 0.930934
\(513\) −6.06758e8 −0.198429
\(514\) 4.37983e9 1.42261
\(515\) 3.97756e9 1.28319
\(516\) 3.70819e8 0.118820
\(517\) 6.17553e9 1.96543
\(518\) −7.43313e9 −2.34973
\(519\) 2.28108e9 0.716233
\(520\) −1.10806e9 −0.345582
\(521\) −4.64043e9 −1.43756 −0.718780 0.695238i \(-0.755299\pi\)
−0.718780 + 0.695238i \(0.755299\pi\)
\(522\) 1.14392e9 0.352005
\(523\) 6.14798e9 1.87922 0.939608 0.342254i \(-0.111190\pi\)
0.939608 + 0.342254i \(0.111190\pi\)
\(524\) −4.22983e9 −1.28429
\(525\) 5.59840e9 1.68852
\(526\) 4.07230e9 1.22008
\(527\) 8.22772e9 2.44874
\(528\) −3.67671e9 −1.08703
\(529\) −7.58342e8 −0.222726
\(530\) −8.66704e9 −2.52875
\(531\) 1.49721e8 0.0433963
\(532\) −4.27662e9 −1.23143
\(533\) −9.38269e8 −0.268400
\(534\) −2.52581e9 −0.717804
\(535\) −4.63585e9 −1.30885
\(536\) 1.60764e9 0.450933
\(537\) −2.13463e8 −0.0594856
\(538\) 3.98429e9 1.10310
\(539\) 1.13678e10 3.12691
\(540\) 7.93137e8 0.216756
\(541\) −1.05096e9 −0.285362 −0.142681 0.989769i \(-0.545572\pi\)
−0.142681 + 0.989769i \(0.545572\pi\)
\(542\) −7.19377e9 −1.94070
\(543\) 1.14293e9 0.306351
\(544\) 7.57407e9 2.01713
\(545\) −2.03615e9 −0.538794
\(546\) 2.57533e9 0.677109
\(547\) 5.58405e9 1.45879 0.729396 0.684092i \(-0.239801\pi\)
0.729396 + 0.684092i \(0.239801\pi\)
\(548\) −1.59872e9 −0.414994
\(549\) 5.24225e8 0.135212
\(550\) 1.32257e10 3.38960
\(551\) −3.29125e9 −0.838167
\(552\) 8.16476e8 0.206612
\(553\) 4.60283e9 1.15741
\(554\) −4.06746e9 −1.01634
\(555\) −3.96629e9 −0.984826
\(556\) 2.31691e9 0.571673
\(557\) −2.32229e7 −0.00569408 −0.00284704 0.999996i \(-0.500906\pi\)
−0.00284704 + 0.999996i \(0.500906\pi\)
\(558\) −2.52896e9 −0.616201
\(559\) 6.42480e8 0.155567
\(560\) −1.43669e10 −3.45703
\(561\) −6.43901e9 −1.53975
\(562\) 4.82174e9 1.14585
\(563\) 4.34065e9 1.02512 0.512561 0.858651i \(-0.328697\pi\)
0.512561 + 0.858651i \(0.328697\pi\)
\(564\) 2.14478e9 0.503391
\(565\) −9.69060e8 −0.226038
\(566\) 3.63722e9 0.843163
\(567\) 8.37775e8 0.193013
\(568\) −1.65671e9 −0.379338
\(569\) −1.25447e9 −0.285474 −0.142737 0.989761i \(-0.545590\pi\)
−0.142737 + 0.989761i \(0.545590\pi\)
\(570\) −5.60109e9 −1.26681
\(571\) −7.73775e9 −1.73936 −0.869678 0.493620i \(-0.835673\pi\)
−0.869678 + 0.493620i \(0.835673\pi\)
\(572\) 2.47872e9 0.553786
\(573\) −2.20259e9 −0.489094
\(574\) −5.28039e9 −1.16540
\(575\) −6.76646e9 −1.48431
\(576\) −4.70784e8 −0.102646
\(577\) 6.30012e9 1.36532 0.682658 0.730738i \(-0.260824\pi\)
0.682658 + 0.730738i \(0.260824\pi\)
\(578\) 1.18268e10 2.54754
\(579\) 9.04005e8 0.193551
\(580\) 4.30224e9 0.915580
\(581\) −1.04358e10 −2.20754
\(582\) 6.88707e9 1.44812
\(583\) −8.81142e9 −1.84164
\(584\) 5.17680e8 0.107551
\(585\) 1.37419e9 0.283792
\(586\) 5.48375e9 1.12573
\(587\) −2.77083e9 −0.565426 −0.282713 0.959204i \(-0.591234\pi\)
−0.282713 + 0.959204i \(0.591234\pi\)
\(588\) 3.94807e9 0.800873
\(589\) 7.27625e9 1.46725
\(590\) 1.38210e9 0.277050
\(591\) −1.53729e9 −0.306338
\(592\) 6.38563e9 1.26496
\(593\) −5.31671e9 −1.04701 −0.523506 0.852022i \(-0.675376\pi\)
−0.523506 + 0.852022i \(0.675376\pi\)
\(594\) 1.97916e9 0.387462
\(595\) −2.51607e10 −4.89680
\(596\) 5.84764e9 1.13141
\(597\) 9.94783e8 0.191346
\(598\) −3.11265e9 −0.595219
\(599\) −5.08449e9 −0.966616 −0.483308 0.875450i \(-0.660565\pi\)
−0.483308 + 0.875450i \(0.660565\pi\)
\(600\) −2.08755e9 −0.394554
\(601\) −5.61219e9 −1.05456 −0.527280 0.849691i \(-0.676788\pi\)
−0.527280 + 0.849691i \(0.676788\pi\)
\(602\) 3.61575e9 0.675477
\(603\) −1.99375e9 −0.370306
\(604\) −6.14935e9 −1.13553
\(605\) 1.25096e10 2.29668
\(606\) 5.87595e9 1.07257
\(607\) 6.49781e9 1.17925 0.589626 0.807677i \(-0.299275\pi\)
0.589626 + 0.807677i \(0.299275\pi\)
\(608\) 6.69819e9 1.20863
\(609\) 4.54437e9 0.815292
\(610\) 4.83921e9 0.863217
\(611\) 3.71604e9 0.659076
\(612\) −2.23629e9 −0.394364
\(613\) 3.54849e9 0.622203 0.311102 0.950377i \(-0.399302\pi\)
0.311102 + 0.950377i \(0.399302\pi\)
\(614\) −7.51762e9 −1.31066
\(615\) −2.81760e9 −0.488446
\(616\) −6.33982e9 −1.09281
\(617\) −8.10427e9 −1.38904 −0.694522 0.719472i \(-0.744384\pi\)
−0.694522 + 0.719472i \(0.744384\pi\)
\(618\) 3.44714e9 0.587488
\(619\) −4.62805e9 −0.784298 −0.392149 0.919902i \(-0.628268\pi\)
−0.392149 + 0.919902i \(0.628268\pi\)
\(620\) −9.51131e9 −1.60276
\(621\) −1.01257e9 −0.169670
\(622\) 8.92240e7 0.0148667
\(623\) −1.00341e10 −1.66253
\(624\) −2.21241e9 −0.364518
\(625\) 9.20980e8 0.150893
\(626\) 1.32417e8 0.0215742
\(627\) −5.69439e9 −0.922594
\(628\) −2.33646e9 −0.376443
\(629\) 1.11831e10 1.79179
\(630\) 7.73365e9 1.23223
\(631\) −2.22522e9 −0.352590 −0.176295 0.984337i \(-0.556411\pi\)
−0.176295 + 0.984337i \(0.556411\pi\)
\(632\) −1.71632e9 −0.270450
\(633\) 1.73551e8 0.0271966
\(634\) 2.26525e9 0.353024
\(635\) 6.55663e9 1.01618
\(636\) −3.06023e9 −0.471687
\(637\) 6.84041e9 1.04856
\(638\) 1.07356e10 1.63665
\(639\) 2.05461e9 0.311513
\(640\) 8.38904e9 1.26498
\(641\) −3.74794e9 −0.562069 −0.281034 0.959698i \(-0.590677\pi\)
−0.281034 + 0.959698i \(0.590677\pi\)
\(642\) −4.01764e9 −0.599237
\(643\) −6.50343e9 −0.964726 −0.482363 0.875971i \(-0.660221\pi\)
−0.482363 + 0.875971i \(0.660221\pi\)
\(644\) −7.13692e9 −1.05296
\(645\) 1.92935e9 0.283108
\(646\) 1.57925e10 2.30482
\(647\) −4.25390e9 −0.617479 −0.308740 0.951147i \(-0.599907\pi\)
−0.308740 + 0.951147i \(0.599907\pi\)
\(648\) −3.12392e8 −0.0451011
\(649\) 1.40513e9 0.201771
\(650\) 7.95836e9 1.13665
\(651\) −1.00466e10 −1.42721
\(652\) 5.06808e9 0.716106
\(653\) 3.65504e9 0.513684 0.256842 0.966453i \(-0.417318\pi\)
0.256842 + 0.966453i \(0.417318\pi\)
\(654\) −1.76462e9 −0.246678
\(655\) −2.20076e10 −3.06005
\(656\) 4.53626e9 0.627386
\(657\) −6.42013e8 −0.0883212
\(658\) 2.09131e10 2.86173
\(659\) −9.45163e9 −1.28649 −0.643247 0.765659i \(-0.722413\pi\)
−0.643247 + 0.765659i \(0.722413\pi\)
\(660\) 7.44355e9 1.00780
\(661\) −4.38475e9 −0.590527 −0.295264 0.955416i \(-0.595407\pi\)
−0.295264 + 0.955416i \(0.595407\pi\)
\(662\) 9.80086e9 1.31299
\(663\) −3.87458e9 −0.516330
\(664\) 3.89132e9 0.515832
\(665\) −2.22510e10 −2.93410
\(666\) −3.43737e9 −0.450886
\(667\) −5.49252e9 −0.716690
\(668\) 4.38701e8 0.0569444
\(669\) −1.60205e9 −0.206864
\(670\) −1.84047e10 −2.36411
\(671\) 4.91982e9 0.628667
\(672\) −9.24846e9 −1.17565
\(673\) −1.06771e10 −1.35020 −0.675102 0.737724i \(-0.735900\pi\)
−0.675102 + 0.737724i \(0.735900\pi\)
\(674\) 1.04658e10 1.31663
\(675\) 2.58892e9 0.324008
\(676\) −4.03060e9 −0.501830
\(677\) 5.43224e8 0.0672851 0.0336426 0.999434i \(-0.489289\pi\)
0.0336426 + 0.999434i \(0.489289\pi\)
\(678\) −8.39832e8 −0.103488
\(679\) 2.73597e10 3.35404
\(680\) 9.38198e9 1.14423
\(681\) 6.74858e9 0.818837
\(682\) −2.37342e10 −2.86503
\(683\) −9.65996e8 −0.116012 −0.0580060 0.998316i \(-0.518474\pi\)
−0.0580060 + 0.998316i \(0.518474\pi\)
\(684\) −1.97768e9 −0.236297
\(685\) −8.31808e9 −0.988794
\(686\) 1.94160e10 2.29628
\(687\) −4.96498e9 −0.584210
\(688\) −3.10621e9 −0.363639
\(689\) −5.30214e9 −0.617567
\(690\) −9.34723e9 −1.08321
\(691\) −6.65558e9 −0.767384 −0.383692 0.923461i \(-0.625348\pi\)
−0.383692 + 0.923461i \(0.625348\pi\)
\(692\) 7.43498e9 0.852920
\(693\) 7.86248e9 0.897415
\(694\) 1.00702e10 1.14362
\(695\) 1.20548e10 1.36211
\(696\) −1.69452e9 −0.190508
\(697\) 7.94435e9 0.888677
\(698\) −6.04004e9 −0.672273
\(699\) −3.62384e9 −0.401328
\(700\) 1.82475e10 2.01076
\(701\) −7.59130e9 −0.832345 −0.416172 0.909286i \(-0.636629\pi\)
−0.416172 + 0.909286i \(0.636629\pi\)
\(702\) 1.19093e9 0.129929
\(703\) 9.88990e9 1.07361
\(704\) −4.41828e9 −0.477254
\(705\) 1.11592e10 1.19942
\(706\) 1.08303e10 1.15830
\(707\) 2.33429e10 2.48421
\(708\) 4.88004e8 0.0516782
\(709\) −7.76529e9 −0.818268 −0.409134 0.912474i \(-0.634169\pi\)
−0.409134 + 0.912474i \(0.634169\pi\)
\(710\) 1.89664e10 1.98876
\(711\) 2.12853e9 0.222094
\(712\) 3.74154e9 0.388481
\(713\) 1.21428e10 1.25460
\(714\) −2.18054e10 −2.24192
\(715\) 1.28967e10 1.31949
\(716\) −6.95763e8 −0.0708380
\(717\) −1.02569e10 −1.03920
\(718\) 2.26054e10 2.27917
\(719\) 2.04713e9 0.205398 0.102699 0.994713i \(-0.467252\pi\)
0.102699 + 0.994713i \(0.467252\pi\)
\(720\) −6.64380e9 −0.663366
\(721\) 1.36942e10 1.36070
\(722\) 8.28926e8 0.0819664
\(723\) −6.70574e8 −0.0659876
\(724\) 3.72528e9 0.364816
\(725\) 1.40431e10 1.36862
\(726\) 1.08414e10 1.05150
\(727\) −7.27546e9 −0.702247 −0.351124 0.936329i \(-0.614200\pi\)
−0.351124 + 0.936329i \(0.614200\pi\)
\(728\) −3.81490e9 −0.366457
\(729\) 3.87420e8 0.0370370
\(730\) −5.92654e9 −0.563860
\(731\) −5.43990e9 −0.515086
\(732\) 1.70867e9 0.161016
\(733\) 1.92809e9 0.180827 0.0904135 0.995904i \(-0.471181\pi\)
0.0904135 + 0.995904i \(0.471181\pi\)
\(734\) −1.46941e10 −1.37153
\(735\) 2.05416e10 1.90822
\(736\) 1.11781e10 1.03346
\(737\) −1.87113e10 −1.72174
\(738\) −2.44186e9 −0.223627
\(739\) −1.32836e9 −0.121077 −0.0605385 0.998166i \(-0.519282\pi\)
−0.0605385 + 0.998166i \(0.519282\pi\)
\(740\) −1.29278e10 −1.17277
\(741\) −3.42652e9 −0.309378
\(742\) −2.98394e10 −2.68149
\(743\) 2.02673e10 1.81273 0.906367 0.422490i \(-0.138844\pi\)
0.906367 + 0.422490i \(0.138844\pi\)
\(744\) 3.74621e9 0.333493
\(745\) 3.04250e10 2.69577
\(746\) −1.35210e10 −1.19240
\(747\) −4.82591e9 −0.423601
\(748\) −2.09874e10 −1.83360
\(749\) −1.59606e10 −1.38791
\(750\) 9.70367e9 0.839889
\(751\) −7.01982e9 −0.604764 −0.302382 0.953187i \(-0.597782\pi\)
−0.302382 + 0.953187i \(0.597782\pi\)
\(752\) −1.79660e10 −1.54059
\(753\) −6.22791e9 −0.531569
\(754\) 6.46001e9 0.548825
\(755\) −3.19947e10 −2.70560
\(756\) 2.73066e9 0.229848
\(757\) −1.56061e10 −1.30755 −0.653774 0.756690i \(-0.726815\pi\)
−0.653774 + 0.756690i \(0.726815\pi\)
\(758\) 1.78664e10 1.49003
\(759\) −9.50293e9 −0.788881
\(760\) 8.29703e9 0.685606
\(761\) 1.06411e8 0.00875264 0.00437632 0.999990i \(-0.498607\pi\)
0.00437632 + 0.999990i \(0.498607\pi\)
\(762\) 5.68227e9 0.465243
\(763\) −7.01018e9 −0.571339
\(764\) −7.17916e9 −0.582434
\(765\) −1.16353e10 −0.939641
\(766\) 5.66685e9 0.455556
\(767\) 8.45515e8 0.0676608
\(768\) 9.50219e9 0.756936
\(769\) 1.40421e9 0.111350 0.0556750 0.998449i \(-0.482269\pi\)
0.0556750 + 0.998449i \(0.482269\pi\)
\(770\) 7.25799e10 5.72926
\(771\) 8.04618e9 0.632266
\(772\) 2.94653e9 0.230489
\(773\) 1.70826e10 1.33022 0.665112 0.746744i \(-0.268384\pi\)
0.665112 + 0.746744i \(0.268384\pi\)
\(774\) 1.67206e9 0.129616
\(775\) −3.10464e10 −2.39582
\(776\) −1.02020e10 −0.783734
\(777\) −1.36554e10 −1.04431
\(778\) −2.04421e10 −1.55631
\(779\) 7.02565e9 0.532483
\(780\) 4.47905e9 0.337952
\(781\) 1.92824e10 1.44838
\(782\) 2.63549e10 1.97078
\(783\) 2.10150e9 0.156445
\(784\) −3.30714e10 −2.45102
\(785\) −1.21565e10 −0.896942
\(786\) −1.90728e10 −1.40099
\(787\) −9.46350e9 −0.692054 −0.346027 0.938224i \(-0.612470\pi\)
−0.346027 + 0.938224i \(0.612470\pi\)
\(788\) −5.01068e9 −0.364800
\(789\) 7.48121e9 0.542254
\(790\) 1.96488e10 1.41789
\(791\) −3.33634e9 −0.239691
\(792\) −2.93178e9 −0.209698
\(793\) 2.96043e9 0.210814
\(794\) −1.97924e9 −0.140322
\(795\) −1.59222e10 −1.12388
\(796\) 3.24241e9 0.227862
\(797\) −1.05899e10 −0.740949 −0.370474 0.928843i \(-0.620805\pi\)
−0.370474 + 0.928843i \(0.620805\pi\)
\(798\) −1.92838e10 −1.34333
\(799\) −3.14638e10 −2.18221
\(800\) −2.85799e10 −1.97354
\(801\) −4.64016e9 −0.319021
\(802\) 1.16435e10 0.797029
\(803\) −6.02526e9 −0.410650
\(804\) −6.49848e9 −0.440977
\(805\) −3.71330e10 −2.50885
\(806\) −1.42817e10 −0.960743
\(807\) 7.31954e9 0.490260
\(808\) −8.70418e9 −0.580481
\(809\) 1.90523e10 1.26511 0.632555 0.774516i \(-0.282006\pi\)
0.632555 + 0.774516i \(0.282006\pi\)
\(810\) 3.57635e9 0.236451
\(811\) 1.62908e10 1.07243 0.536217 0.844080i \(-0.319853\pi\)
0.536217 + 0.844080i \(0.319853\pi\)
\(812\) 1.48120e10 0.970884
\(813\) −1.32157e10 −0.862527
\(814\) −3.22595e10 −2.09639
\(815\) 2.63690e10 1.70625
\(816\) 1.87325e10 1.20693
\(817\) −4.81081e9 −0.308632
\(818\) −2.91664e10 −1.86314
\(819\) 4.73113e9 0.300934
\(820\) −9.18374e9 −0.581662
\(821\) 1.60953e9 0.101508 0.0507539 0.998711i \(-0.483838\pi\)
0.0507539 + 0.998711i \(0.483838\pi\)
\(822\) −7.20883e9 −0.452703
\(823\) −1.20942e10 −0.756272 −0.378136 0.925750i \(-0.623435\pi\)
−0.378136 + 0.925750i \(0.623435\pi\)
\(824\) −5.10633e9 −0.317953
\(825\) 2.42969e10 1.50647
\(826\) 4.75839e9 0.293785
\(827\) 9.39665e9 0.577702 0.288851 0.957374i \(-0.406727\pi\)
0.288851 + 0.957374i \(0.406727\pi\)
\(828\) −3.30039e9 −0.202050
\(829\) 1.58388e10 0.965563 0.482781 0.875741i \(-0.339627\pi\)
0.482781 + 0.875741i \(0.339627\pi\)
\(830\) −4.45488e10 −2.70435
\(831\) −7.47232e9 −0.451702
\(832\) −2.65864e9 −0.160040
\(833\) −5.79180e10 −3.47181
\(834\) 1.04472e10 0.623619
\(835\) 2.28254e9 0.135680
\(836\) −1.85604e10 −1.09866
\(837\) −4.64595e9 −0.273864
\(838\) 1.64622e10 0.966346
\(839\) 3.24355e10 1.89607 0.948034 0.318170i \(-0.103068\pi\)
0.948034 + 0.318170i \(0.103068\pi\)
\(840\) −1.14560e10 −0.666895
\(841\) −5.85067e9 −0.339172
\(842\) 2.35380e10 1.35887
\(843\) 8.85801e9 0.509260
\(844\) 5.65676e8 0.0323869
\(845\) −2.09710e10 −1.19570
\(846\) 9.67105e9 0.549133
\(847\) 4.30689e10 2.43541
\(848\) 2.56343e10 1.44357
\(849\) 6.68193e9 0.374736
\(850\) −6.73837e10 −3.76347
\(851\) 1.65045e10 0.918013
\(852\) 6.69682e9 0.370962
\(853\) 4.28375e9 0.236321 0.118161 0.992995i \(-0.462300\pi\)
0.118161 + 0.992995i \(0.462300\pi\)
\(854\) 1.66607e10 0.915358
\(855\) −1.02898e10 −0.563020
\(856\) 5.95142e9 0.324312
\(857\) −2.60684e9 −0.141475 −0.0707377 0.997495i \(-0.522535\pi\)
−0.0707377 + 0.997495i \(0.522535\pi\)
\(858\) 1.11768e10 0.604107
\(859\) 7.87750e8 0.0424046 0.0212023 0.999775i \(-0.493251\pi\)
0.0212023 + 0.999775i \(0.493251\pi\)
\(860\) 6.28856e9 0.337137
\(861\) −9.70060e9 −0.517950
\(862\) −2.09024e10 −1.11153
\(863\) −1.69537e10 −0.897896 −0.448948 0.893558i \(-0.648201\pi\)
−0.448948 + 0.893558i \(0.648201\pi\)
\(864\) −4.27685e9 −0.225593
\(865\) 3.86838e10 2.03223
\(866\) 1.66523e10 0.871290
\(867\) 2.17271e10 1.13223
\(868\) −3.27461e10 −1.69958
\(869\) 1.99761e10 1.03262
\(870\) 1.93993e10 0.998776
\(871\) −1.12592e10 −0.577359
\(872\) 2.61398e9 0.133504
\(873\) 1.26522e10 0.643602
\(874\) 2.33072e10 1.18086
\(875\) 3.85491e10 1.94530
\(876\) −2.09259e9 −0.105177
\(877\) −1.67825e10 −0.840153 −0.420076 0.907489i \(-0.637997\pi\)
−0.420076 + 0.907489i \(0.637997\pi\)
\(878\) −2.83562e10 −1.41390
\(879\) 1.00742e10 0.500321
\(880\) −6.23517e10 −3.08432
\(881\) 3.74442e10 1.84488 0.922442 0.386135i \(-0.126190\pi\)
0.922442 + 0.386135i \(0.126190\pi\)
\(882\) 1.78023e10 0.873647
\(883\) 2.15204e10 1.05193 0.525967 0.850505i \(-0.323703\pi\)
0.525967 + 0.850505i \(0.323703\pi\)
\(884\) −1.26289e10 −0.614868
\(885\) 2.53906e9 0.123132
\(886\) −1.22531e9 −0.0591873
\(887\) −1.11581e10 −0.536857 −0.268429 0.963300i \(-0.586504\pi\)
−0.268429 + 0.963300i \(0.586504\pi\)
\(888\) 5.09186e9 0.244023
\(889\) 2.25735e10 1.07756
\(890\) −4.28341e10 −2.03669
\(891\) 3.63592e9 0.172204
\(892\) −5.22174e9 −0.246342
\(893\) −2.78252e10 −1.30755
\(894\) 2.63677e10 1.23421
\(895\) −3.62002e9 −0.168784
\(896\) 2.88823e10 1.34138
\(897\) −5.71825e9 −0.264539
\(898\) 4.24596e10 1.95663
\(899\) −2.52012e10 −1.15681
\(900\) 8.43837e9 0.385842
\(901\) 4.48934e10 2.04478
\(902\) −2.29167e10 −1.03975
\(903\) 6.64249e9 0.300209
\(904\) 1.24406e9 0.0560083
\(905\) 1.93825e10 0.869238
\(906\) −2.77281e10 −1.23872
\(907\) 7.89855e9 0.351497 0.175749 0.984435i \(-0.443765\pi\)
0.175749 + 0.984435i \(0.443765\pi\)
\(908\) 2.19964e10 0.975106
\(909\) 1.07947e10 0.476691
\(910\) 4.36739e10 1.92122
\(911\) 1.68370e10 0.737821 0.368911 0.929465i \(-0.379731\pi\)
0.368911 + 0.929465i \(0.379731\pi\)
\(912\) 1.65662e10 0.723172
\(913\) −4.52909e10 −1.96953
\(914\) −3.28621e10 −1.42359
\(915\) 8.89011e9 0.383648
\(916\) −1.61829e10 −0.695702
\(917\) −7.57691e10 −3.24489
\(918\) −1.00837e10 −0.430199
\(919\) 2.78362e10 1.18306 0.591529 0.806284i \(-0.298525\pi\)
0.591529 + 0.806284i \(0.298525\pi\)
\(920\) 1.38463e10 0.586240
\(921\) −1.38106e10 −0.582511
\(922\) −1.85830e10 −0.780830
\(923\) 1.16029e10 0.485691
\(924\) 2.56271e10 1.06868
\(925\) −4.21983e10 −1.75307
\(926\) 3.51572e10 1.45504
\(927\) 6.33273e9 0.261103
\(928\) −2.31991e10 −0.952911
\(929\) 2.25578e10 0.923083 0.461542 0.887119i \(-0.347296\pi\)
0.461542 + 0.887119i \(0.347296\pi\)
\(930\) −4.28876e10 −1.74840
\(931\) −5.12202e10 −2.08026
\(932\) −1.18116e10 −0.477918
\(933\) 1.63913e8 0.00660737
\(934\) 8.57279e9 0.344277
\(935\) −1.09197e11 −4.36886
\(936\) −1.76416e9 −0.0703189
\(937\) −2.56679e10 −1.01930 −0.509649 0.860382i \(-0.670225\pi\)
−0.509649 + 0.860382i \(0.670225\pi\)
\(938\) −6.33648e10 −2.50691
\(939\) 2.43263e8 0.00958842
\(940\) 3.63724e10 1.42832
\(941\) 7.37013e9 0.288344 0.144172 0.989553i \(-0.453948\pi\)
0.144172 + 0.989553i \(0.453948\pi\)
\(942\) −1.05354e10 −0.410650
\(943\) 1.17246e10 0.455309
\(944\) −4.08782e9 −0.158158
\(945\) 1.42075e10 0.547654
\(946\) 1.56922e10 0.602651
\(947\) 4.90023e10 1.87496 0.937479 0.348042i \(-0.113153\pi\)
0.937479 + 0.348042i \(0.113153\pi\)
\(948\) 6.93776e9 0.264478
\(949\) −3.62561e9 −0.137705
\(950\) −5.95912e10 −2.25502
\(951\) 4.16149e9 0.156898
\(952\) 3.23008e10 1.21335
\(953\) −2.07767e10 −0.777590 −0.388795 0.921324i \(-0.627109\pi\)
−0.388795 + 0.921324i \(0.627109\pi\)
\(954\) −1.37989e10 −0.514548
\(955\) −3.73528e10 −1.38775
\(956\) −3.34315e10 −1.23752
\(957\) 1.97224e10 0.727392
\(958\) 5.57557e10 2.04885
\(959\) −2.86380e10 −1.04852
\(960\) −7.98383e9 −0.291247
\(961\) 2.82017e10 1.02505
\(962\) −1.94117e10 −0.702993
\(963\) −7.38080e9 −0.266325
\(964\) −2.18568e9 −0.0785809
\(965\) 1.53306e10 0.549180
\(966\) −3.21812e10 −1.14864
\(967\) −1.99822e10 −0.710642 −0.355321 0.934744i \(-0.615628\pi\)
−0.355321 + 0.934744i \(0.615628\pi\)
\(968\) −1.60597e10 −0.569079
\(969\) 2.90124e10 1.02436
\(970\) 1.16795e11 4.10888
\(971\) −1.36752e9 −0.0479365 −0.0239683 0.999713i \(-0.507630\pi\)
−0.0239683 + 0.999713i \(0.507630\pi\)
\(972\) 1.26276e9 0.0441053
\(973\) 4.15029e10 1.44439
\(974\) −3.52681e10 −1.22300
\(975\) 1.46203e10 0.505173
\(976\) −1.43128e10 −0.492778
\(977\) −2.85359e10 −0.978951 −0.489475 0.872017i \(-0.662812\pi\)
−0.489475 + 0.872017i \(0.662812\pi\)
\(978\) 2.28526e10 0.781176
\(979\) −4.35476e10 −1.48329
\(980\) 6.69536e10 2.27239
\(981\) −3.24179e9 −0.109633
\(982\) 5.89725e10 1.98728
\(983\) 1.22389e10 0.410965 0.205483 0.978661i \(-0.434124\pi\)
0.205483 + 0.978661i \(0.434124\pi\)
\(984\) 3.61719e9 0.121029
\(985\) −2.60703e10 −0.869200
\(986\) −5.46971e10 −1.81717
\(987\) 3.84195e10 1.27187
\(988\) −1.11684e10 −0.368420
\(989\) −8.02840e9 −0.263902
\(990\) 3.35638e10 1.09938
\(991\) −3.67813e10 −1.20052 −0.600260 0.799805i \(-0.704936\pi\)
−0.600260 + 0.799805i \(0.704936\pi\)
\(992\) 5.12881e10 1.66811
\(993\) 1.80052e10 0.583545
\(994\) 6.52987e10 2.10888
\(995\) 1.68701e10 0.542922
\(996\) −1.57296e10 −0.504442
\(997\) −6.04917e10 −1.93314 −0.966569 0.256405i \(-0.917462\pi\)
−0.966569 + 0.256405i \(0.917462\pi\)
\(998\) 3.80370e10 1.21129
\(999\) −6.31479e9 −0.200392
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.1.14 18
3.2 odd 2 531.8.a.e.1.5 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.14 18 1.1 even 1 trivial
531.8.a.e.1.5 18 3.2 odd 2