Properties

Label 177.8.a.d.1.11
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 1.11
Root \(6.43791\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+7.43791 q^{2} +27.0000 q^{3} -72.6775 q^{4} +171.827 q^{5} +200.824 q^{6} +1529.87 q^{7} -1492.62 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+7.43791 q^{2} +27.0000 q^{3} -72.6775 q^{4} +171.827 q^{5} +200.824 q^{6} +1529.87 q^{7} -1492.62 q^{8} +729.000 q^{9} +1278.03 q^{10} -1638.92 q^{11} -1962.29 q^{12} -7315.98 q^{13} +11379.0 q^{14} +4639.33 q^{15} -1799.27 q^{16} +33214.7 q^{17} +5422.24 q^{18} +55570.3 q^{19} -12488.0 q^{20} +41306.4 q^{21} -12190.1 q^{22} -51886.9 q^{23} -40300.8 q^{24} -48600.5 q^{25} -54415.6 q^{26} +19683.0 q^{27} -111187. q^{28} +10383.5 q^{29} +34506.9 q^{30} +92770.0 q^{31} +177673. q^{32} -44250.7 q^{33} +247048. q^{34} +262873. q^{35} -52981.9 q^{36} -504232. q^{37} +413327. q^{38} -197531. q^{39} -256473. q^{40} +635489. q^{41} +307234. q^{42} +101770. q^{43} +119112. q^{44} +125262. q^{45} -385930. q^{46} -72413.1 q^{47} -48580.2 q^{48} +1.51695e6 q^{49} -361486. q^{50} +896798. q^{51} +531707. q^{52} +1.67234e6 q^{53} +146400. q^{54} -281610. q^{55} -2.28351e6 q^{56} +1.50040e6 q^{57} +77231.8 q^{58} +205379. q^{59} -337175. q^{60} +727353. q^{61} +690015. q^{62} +1.11527e6 q^{63} +1.55182e6 q^{64} -1.25708e6 q^{65} -329133. q^{66} +2.81505e6 q^{67} -2.41396e6 q^{68} -1.40095e6 q^{69} +1.95522e6 q^{70} -850222. q^{71} -1.08812e6 q^{72} +5.84245e6 q^{73} -3.75043e6 q^{74} -1.31221e6 q^{75} -4.03871e6 q^{76} -2.50732e6 q^{77} -1.46922e6 q^{78} +2.74755e6 q^{79} -309162. q^{80} +531441. q^{81} +4.72671e6 q^{82} +2.78632e6 q^{83} -3.00205e6 q^{84} +5.70719e6 q^{85} +756957. q^{86} +280355. q^{87} +2.44628e6 q^{88} -6.16092e6 q^{89} +931687. q^{90} -1.11925e7 q^{91} +3.77101e6 q^{92} +2.50479e6 q^{93} -538602. q^{94} +9.54849e6 q^{95} +4.79716e6 q^{96} -1.67686e7 q^{97} +1.12830e7 q^{98} -1.19477e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.43791 0.657425 0.328712 0.944430i \(-0.393385\pi\)
0.328712 + 0.944430i \(0.393385\pi\)
\(3\) 27.0000 0.577350
\(4\) −72.6775 −0.567793
\(5\) 171.827 0.614747 0.307374 0.951589i \(-0.400550\pi\)
0.307374 + 0.951589i \(0.400550\pi\)
\(6\) 200.824 0.379564
\(7\) 1529.87 1.68582 0.842909 0.538056i \(-0.180841\pi\)
0.842909 + 0.538056i \(0.180841\pi\)
\(8\) −1492.62 −1.03071
\(9\) 729.000 0.333333
\(10\) 1278.03 0.404150
\(11\) −1638.92 −0.371263 −0.185632 0.982619i \(-0.559433\pi\)
−0.185632 + 0.982619i \(0.559433\pi\)
\(12\) −1962.29 −0.327815
\(13\) −7315.98 −0.923572 −0.461786 0.886991i \(-0.652791\pi\)
−0.461786 + 0.886991i \(0.652791\pi\)
\(14\) 11379.0 1.10830
\(15\) 4639.33 0.354924
\(16\) −1799.27 −0.109818
\(17\) 33214.7 1.63968 0.819841 0.572592i \(-0.194062\pi\)
0.819841 + 0.572592i \(0.194062\pi\)
\(18\) 5422.24 0.219142
\(19\) 55570.3 1.85868 0.929342 0.369221i \(-0.120375\pi\)
0.929342 + 0.369221i \(0.120375\pi\)
\(20\) −12488.0 −0.349049
\(21\) 41306.4 0.973308
\(22\) −12190.1 −0.244078
\(23\) −51886.9 −0.889222 −0.444611 0.895724i \(-0.646658\pi\)
−0.444611 + 0.895724i \(0.646658\pi\)
\(24\) −40300.8 −0.595078
\(25\) −48600.5 −0.622086
\(26\) −54415.6 −0.607179
\(27\) 19683.0 0.192450
\(28\) −111187. −0.957196
\(29\) 10383.5 0.0790591 0.0395296 0.999218i \(-0.487414\pi\)
0.0395296 + 0.999218i \(0.487414\pi\)
\(30\) 34506.9 0.233336
\(31\) 92770.0 0.559296 0.279648 0.960103i \(-0.409782\pi\)
0.279648 + 0.960103i \(0.409782\pi\)
\(32\) 177673. 0.958508
\(33\) −44250.7 −0.214349
\(34\) 247048. 1.07797
\(35\) 262873. 1.03635
\(36\) −52981.9 −0.189264
\(37\) −504232. −1.63653 −0.818266 0.574840i \(-0.805064\pi\)
−0.818266 + 0.574840i \(0.805064\pi\)
\(38\) 413327. 1.22194
\(39\) −197531. −0.533224
\(40\) −256473. −0.633623
\(41\) 635489. 1.44001 0.720003 0.693971i \(-0.244140\pi\)
0.720003 + 0.693971i \(0.244140\pi\)
\(42\) 307234. 0.639877
\(43\) 101770. 0.195200 0.0976002 0.995226i \(-0.468883\pi\)
0.0976002 + 0.995226i \(0.468883\pi\)
\(44\) 119112. 0.210801
\(45\) 125262. 0.204916
\(46\) −385930. −0.584597
\(47\) −72413.1 −0.101736 −0.0508680 0.998705i \(-0.516199\pi\)
−0.0508680 + 0.998705i \(0.516199\pi\)
\(48\) −48580.2 −0.0634037
\(49\) 1.51695e6 1.84198
\(50\) −361486. −0.408975
\(51\) 896798. 0.946671
\(52\) 531707. 0.524397
\(53\) 1.67234e6 1.54297 0.771487 0.636245i \(-0.219513\pi\)
0.771487 + 0.636245i \(0.219513\pi\)
\(54\) 146400. 0.126521
\(55\) −281610. −0.228233
\(56\) −2.28351e6 −1.73758
\(57\) 1.50040e6 1.07311
\(58\) 77231.8 0.0519754
\(59\) 205379. 0.130189
\(60\) −337175. −0.201524
\(61\) 727353. 0.410290 0.205145 0.978732i \(-0.434233\pi\)
0.205145 + 0.978732i \(0.434233\pi\)
\(62\) 690015. 0.367695
\(63\) 1.11527e6 0.561940
\(64\) 1.55182e6 0.739965
\(65\) −1.25708e6 −0.567763
\(66\) −329133. −0.140918
\(67\) 2.81505e6 1.14347 0.571735 0.820439i \(-0.306271\pi\)
0.571735 + 0.820439i \(0.306271\pi\)
\(68\) −2.41396e6 −0.930999
\(69\) −1.40095e6 −0.513393
\(70\) 1.95522e6 0.681323
\(71\) −850222. −0.281921 −0.140961 0.990015i \(-0.545019\pi\)
−0.140961 + 0.990015i \(0.545019\pi\)
\(72\) −1.08812e6 −0.343569
\(73\) 5.84245e6 1.75778 0.878891 0.477023i \(-0.158284\pi\)
0.878891 + 0.477023i \(0.158284\pi\)
\(74\) −3.75043e6 −1.07590
\(75\) −1.31221e6 −0.359161
\(76\) −4.03871e6 −1.05535
\(77\) −2.50732e6 −0.625883
\(78\) −1.46922e6 −0.350555
\(79\) 2.74755e6 0.626977 0.313488 0.949592i \(-0.398502\pi\)
0.313488 + 0.949592i \(0.398502\pi\)
\(80\) −309162. −0.0675106
\(81\) 531441. 0.111111
\(82\) 4.72671e6 0.946695
\(83\) 2.78632e6 0.534882 0.267441 0.963574i \(-0.413822\pi\)
0.267441 + 0.963574i \(0.413822\pi\)
\(84\) −3.00205e6 −0.552637
\(85\) 5.70719e6 1.00799
\(86\) 756957. 0.128330
\(87\) 280355. 0.0456448
\(88\) 2.44628e6 0.382663
\(89\) −6.16092e6 −0.926361 −0.463181 0.886264i \(-0.653292\pi\)
−0.463181 + 0.886264i \(0.653292\pi\)
\(90\) 931687. 0.134717
\(91\) −1.11925e7 −1.55697
\(92\) 3.77101e6 0.504894
\(93\) 2.50479e6 0.322910
\(94\) −538602. −0.0668838
\(95\) 9.54849e6 1.14262
\(96\) 4.79716e6 0.553395
\(97\) −1.67686e7 −1.86550 −0.932751 0.360521i \(-0.882599\pi\)
−0.932751 + 0.360521i \(0.882599\pi\)
\(98\) 1.12830e7 1.21097
\(99\) −1.19477e6 −0.123754
\(100\) 3.53216e6 0.353216
\(101\) 7.93944e6 0.766770 0.383385 0.923589i \(-0.374758\pi\)
0.383385 + 0.923589i \(0.374758\pi\)
\(102\) 6.67030e6 0.622365
\(103\) 1.76750e7 1.59378 0.796889 0.604125i \(-0.206477\pi\)
0.796889 + 0.604125i \(0.206477\pi\)
\(104\) 1.09200e7 0.951931
\(105\) 7.09756e6 0.598338
\(106\) 1.24387e7 1.01439
\(107\) −9.16951e6 −0.723607 −0.361803 0.932254i \(-0.617839\pi\)
−0.361803 + 0.932254i \(0.617839\pi\)
\(108\) −1.43051e6 −0.109272
\(109\) 1.32228e7 0.977982 0.488991 0.872289i \(-0.337365\pi\)
0.488991 + 0.872289i \(0.337365\pi\)
\(110\) −2.09459e6 −0.150046
\(111\) −1.36143e7 −0.944852
\(112\) −2.75264e6 −0.185134
\(113\) −2.66031e7 −1.73443 −0.867216 0.497932i \(-0.834093\pi\)
−0.867216 + 0.497932i \(0.834093\pi\)
\(114\) 1.11598e7 0.705490
\(115\) −8.91557e6 −0.546647
\(116\) −754649. −0.0448892
\(117\) −5.33335e6 −0.307857
\(118\) 1.52759e6 0.0855894
\(119\) 5.08142e7 2.76421
\(120\) −6.92476e6 −0.365823
\(121\) −1.68011e7 −0.862163
\(122\) 5.40999e6 0.269735
\(123\) 1.71582e7 0.831388
\(124\) −6.74229e6 −0.317564
\(125\) −2.17749e7 −0.997173
\(126\) 8.29531e6 0.369433
\(127\) −2.04342e7 −0.885209 −0.442604 0.896717i \(-0.645945\pi\)
−0.442604 + 0.896717i \(0.645945\pi\)
\(128\) −1.11998e7 −0.472037
\(129\) 2.74779e6 0.112699
\(130\) −9.35007e6 −0.373261
\(131\) −4.18739e7 −1.62740 −0.813699 0.581286i \(-0.802550\pi\)
−0.813699 + 0.581286i \(0.802550\pi\)
\(132\) 3.21603e6 0.121706
\(133\) 8.50153e7 3.13340
\(134\) 2.09381e7 0.751745
\(135\) 3.38207e6 0.118308
\(136\) −4.95770e7 −1.69003
\(137\) 5.60709e7 1.86301 0.931506 0.363725i \(-0.118495\pi\)
0.931506 + 0.363725i \(0.118495\pi\)
\(138\) −1.04201e7 −0.337517
\(139\) −3.90420e7 −1.23305 −0.616524 0.787336i \(-0.711460\pi\)
−0.616524 + 0.787336i \(0.711460\pi\)
\(140\) −1.91049e7 −0.588433
\(141\) −1.95515e6 −0.0587373
\(142\) −6.32387e6 −0.185342
\(143\) 1.19903e7 0.342888
\(144\) −1.31166e6 −0.0366062
\(145\) 1.78417e6 0.0486014
\(146\) 4.34556e7 1.15561
\(147\) 4.09577e7 1.06347
\(148\) 3.66463e7 0.929211
\(149\) 5.58520e7 1.38320 0.691602 0.722278i \(-0.256905\pi\)
0.691602 + 0.722278i \(0.256905\pi\)
\(150\) −9.76012e6 −0.236122
\(151\) −6.78120e7 −1.60283 −0.801414 0.598110i \(-0.795919\pi\)
−0.801414 + 0.598110i \(0.795919\pi\)
\(152\) −8.29455e7 −1.91576
\(153\) 2.42135e7 0.546561
\(154\) −1.86493e7 −0.411471
\(155\) 1.59404e7 0.343826
\(156\) 1.43561e7 0.302761
\(157\) 4.45770e7 0.919310 0.459655 0.888098i \(-0.347973\pi\)
0.459655 + 0.888098i \(0.347973\pi\)
\(158\) 2.04361e7 0.412190
\(159\) 4.51532e7 0.890837
\(160\) 3.05290e7 0.589240
\(161\) −7.93801e7 −1.49907
\(162\) 3.95281e6 0.0730472
\(163\) −9.62679e7 −1.74111 −0.870553 0.492075i \(-0.836238\pi\)
−0.870553 + 0.492075i \(0.836238\pi\)
\(164\) −4.61857e7 −0.817625
\(165\) −7.60347e6 −0.131770
\(166\) 2.07244e7 0.351645
\(167\) −3.06157e7 −0.508671 −0.254335 0.967116i \(-0.581857\pi\)
−0.254335 + 0.967116i \(0.581857\pi\)
\(168\) −6.16549e7 −1.00319
\(169\) −9.22500e6 −0.147015
\(170\) 4.24496e7 0.662677
\(171\) 4.05108e7 0.619561
\(172\) −7.39640e6 −0.110833
\(173\) −1.60370e7 −0.235484 −0.117742 0.993044i \(-0.537566\pi\)
−0.117742 + 0.993044i \(0.537566\pi\)
\(174\) 2.08526e6 0.0300080
\(175\) −7.43523e7 −1.04872
\(176\) 2.94884e6 0.0407716
\(177\) 5.54523e6 0.0751646
\(178\) −4.58244e7 −0.609013
\(179\) −2.33939e7 −0.304872 −0.152436 0.988313i \(-0.548712\pi\)
−0.152436 + 0.988313i \(0.548712\pi\)
\(180\) −9.10372e6 −0.116350
\(181\) −9.84290e7 −1.23381 −0.616904 0.787038i \(-0.711614\pi\)
−0.616904 + 0.787038i \(0.711614\pi\)
\(182\) −8.32487e7 −1.02359
\(183\) 1.96385e7 0.236881
\(184\) 7.74475e7 0.916527
\(185\) −8.66407e7 −1.00605
\(186\) 1.86304e7 0.212289
\(187\) −5.44361e7 −0.608754
\(188\) 5.26280e6 0.0577650
\(189\) 3.01124e7 0.324436
\(190\) 7.10208e7 0.751187
\(191\) 3.21552e7 0.333914 0.166957 0.985964i \(-0.446606\pi\)
0.166957 + 0.985964i \(0.446606\pi\)
\(192\) 4.18991e7 0.427219
\(193\) 2.59381e7 0.259709 0.129854 0.991533i \(-0.458549\pi\)
0.129854 + 0.991533i \(0.458549\pi\)
\(194\) −1.24723e8 −1.22643
\(195\) −3.39412e7 −0.327798
\(196\) −1.10248e8 −1.04587
\(197\) −1.60676e7 −0.149733 −0.0748666 0.997194i \(-0.523853\pi\)
−0.0748666 + 0.997194i \(0.523853\pi\)
\(198\) −8.88659e6 −0.0813592
\(199\) 6.47444e7 0.582394 0.291197 0.956663i \(-0.405947\pi\)
0.291197 + 0.956663i \(0.405947\pi\)
\(200\) 7.25421e7 0.641187
\(201\) 7.60064e7 0.660182
\(202\) 5.90529e7 0.504094
\(203\) 1.58854e7 0.133279
\(204\) −6.51770e7 −0.537513
\(205\) 1.09194e8 0.885240
\(206\) 1.31465e8 1.04779
\(207\) −3.78256e7 −0.296407
\(208\) 1.31634e7 0.101425
\(209\) −9.10751e7 −0.690061
\(210\) 5.27910e7 0.393362
\(211\) −4.05485e7 −0.297157 −0.148578 0.988901i \(-0.547470\pi\)
−0.148578 + 0.988901i \(0.547470\pi\)
\(212\) −1.21541e8 −0.876090
\(213\) −2.29560e7 −0.162767
\(214\) −6.82020e7 −0.475717
\(215\) 1.74869e7 0.119999
\(216\) −2.93793e7 −0.198359
\(217\) 1.41926e8 0.942872
\(218\) 9.83501e7 0.642950
\(219\) 1.57746e8 1.01486
\(220\) 2.04667e7 0.129589
\(221\) −2.42998e8 −1.51436
\(222\) −1.01262e8 −0.621169
\(223\) −6.68937e7 −0.403941 −0.201971 0.979392i \(-0.564735\pi\)
−0.201971 + 0.979392i \(0.564735\pi\)
\(224\) 2.71816e8 1.61587
\(225\) −3.54297e7 −0.207362
\(226\) −1.97871e8 −1.14026
\(227\) 1.22298e8 0.693949 0.346975 0.937875i \(-0.387209\pi\)
0.346975 + 0.937875i \(0.387209\pi\)
\(228\) −1.09045e8 −0.609305
\(229\) −2.32389e8 −1.27876 −0.639382 0.768889i \(-0.720810\pi\)
−0.639382 + 0.768889i \(0.720810\pi\)
\(230\) −6.63132e7 −0.359379
\(231\) −6.76978e7 −0.361354
\(232\) −1.54987e7 −0.0814867
\(233\) 7.69697e7 0.398634 0.199317 0.979935i \(-0.436128\pi\)
0.199317 + 0.979935i \(0.436128\pi\)
\(234\) −3.96690e7 −0.202393
\(235\) −1.24425e7 −0.0625419
\(236\) −1.49264e7 −0.0739203
\(237\) 7.41839e7 0.361985
\(238\) 3.77951e8 1.81726
\(239\) −1.06849e7 −0.0506264 −0.0253132 0.999680i \(-0.508058\pi\)
−0.0253132 + 0.999680i \(0.508058\pi\)
\(240\) −8.34739e6 −0.0389773
\(241\) −3.74132e8 −1.72173 −0.860866 0.508831i \(-0.830078\pi\)
−0.860866 + 0.508831i \(0.830078\pi\)
\(242\) −1.24965e8 −0.566808
\(243\) 1.43489e7 0.0641500
\(244\) −5.28622e7 −0.232960
\(245\) 2.60654e8 1.13235
\(246\) 1.27621e8 0.546575
\(247\) −4.06551e8 −1.71663
\(248\) −1.38470e8 −0.576470
\(249\) 7.52307e7 0.308814
\(250\) −1.61959e8 −0.655566
\(251\) 2.63434e8 1.05151 0.525755 0.850636i \(-0.323783\pi\)
0.525755 + 0.850636i \(0.323783\pi\)
\(252\) −8.10553e7 −0.319065
\(253\) 8.50383e7 0.330136
\(254\) −1.51988e8 −0.581958
\(255\) 1.54094e8 0.581963
\(256\) −2.81936e8 −1.05029
\(257\) −3.92808e8 −1.44349 −0.721746 0.692158i \(-0.756660\pi\)
−0.721746 + 0.692158i \(0.756660\pi\)
\(258\) 2.04378e7 0.0740911
\(259\) −7.71409e8 −2.75890
\(260\) 9.13616e7 0.322372
\(261\) 7.56959e6 0.0263530
\(262\) −3.11454e8 −1.06989
\(263\) −2.41410e8 −0.818297 −0.409148 0.912468i \(-0.634174\pi\)
−0.409148 + 0.912468i \(0.634174\pi\)
\(264\) 6.60496e7 0.220931
\(265\) 2.87353e8 0.948539
\(266\) 6.32336e8 2.05998
\(267\) −1.66345e8 −0.534835
\(268\) −2.04591e8 −0.649254
\(269\) 2.35755e8 0.738463 0.369231 0.929337i \(-0.379621\pi\)
0.369231 + 0.929337i \(0.379621\pi\)
\(270\) 2.51555e7 0.0777787
\(271\) −1.49957e8 −0.457692 −0.228846 0.973463i \(-0.573495\pi\)
−0.228846 + 0.973463i \(0.573495\pi\)
\(272\) −5.97621e7 −0.180067
\(273\) −3.02197e8 −0.898920
\(274\) 4.17051e8 1.22479
\(275\) 7.96521e7 0.230958
\(276\) 1.01817e8 0.291501
\(277\) −9.52984e7 −0.269406 −0.134703 0.990886i \(-0.543008\pi\)
−0.134703 + 0.990886i \(0.543008\pi\)
\(278\) −2.90391e8 −0.810637
\(279\) 6.76293e7 0.186432
\(280\) −3.92369e8 −1.06817
\(281\) −7.30263e7 −0.196339 −0.0981696 0.995170i \(-0.531299\pi\)
−0.0981696 + 0.995170i \(0.531299\pi\)
\(282\) −1.45423e7 −0.0386154
\(283\) 1.79740e8 0.471404 0.235702 0.971825i \(-0.424261\pi\)
0.235702 + 0.971825i \(0.424261\pi\)
\(284\) 6.17920e7 0.160073
\(285\) 2.57809e8 0.659692
\(286\) 8.91825e7 0.225423
\(287\) 9.72214e8 2.42759
\(288\) 1.29523e8 0.319503
\(289\) 6.92880e8 1.68856
\(290\) 1.32705e7 0.0319517
\(291\) −4.52752e8 −1.07705
\(292\) −4.24614e8 −0.998056
\(293\) 8.12673e8 1.88747 0.943733 0.330709i \(-0.107288\pi\)
0.943733 + 0.330709i \(0.107288\pi\)
\(294\) 3.04640e8 0.699152
\(295\) 3.52897e7 0.0800333
\(296\) 7.52628e8 1.68678
\(297\) −3.22588e7 −0.0714497
\(298\) 4.15422e8 0.909353
\(299\) 3.79603e8 0.821261
\(300\) 9.53683e7 0.203929
\(301\) 1.55695e8 0.329073
\(302\) −5.04379e8 −1.05374
\(303\) 2.14365e8 0.442695
\(304\) −9.99858e7 −0.204118
\(305\) 1.24979e8 0.252225
\(306\) 1.80098e8 0.359322
\(307\) −1.93647e8 −0.381968 −0.190984 0.981593i \(-0.561168\pi\)
−0.190984 + 0.981593i \(0.561168\pi\)
\(308\) 1.82226e8 0.355372
\(309\) 4.77224e8 0.920168
\(310\) 1.18563e8 0.226039
\(311\) 1.76764e8 0.333220 0.166610 0.986023i \(-0.446718\pi\)
0.166610 + 0.986023i \(0.446718\pi\)
\(312\) 2.94840e8 0.549597
\(313\) 1.36535e8 0.251675 0.125837 0.992051i \(-0.459838\pi\)
0.125837 + 0.992051i \(0.459838\pi\)
\(314\) 3.31560e8 0.604377
\(315\) 1.91634e8 0.345451
\(316\) −1.99685e8 −0.355993
\(317\) −1.67078e8 −0.294586 −0.147293 0.989093i \(-0.547056\pi\)
−0.147293 + 0.989093i \(0.547056\pi\)
\(318\) 3.35845e8 0.585658
\(319\) −1.70177e7 −0.0293518
\(320\) 2.66645e8 0.454892
\(321\) −2.47577e8 −0.417775
\(322\) −5.90422e8 −0.985524
\(323\) 1.84575e9 3.04765
\(324\) −3.86238e7 −0.0630881
\(325\) 3.55560e8 0.574541
\(326\) −7.16032e8 −1.14465
\(327\) 3.57016e8 0.564638
\(328\) −9.48544e8 −1.48422
\(329\) −1.10783e8 −0.171508
\(330\) −5.65539e7 −0.0866291
\(331\) 3.82887e8 0.580327 0.290163 0.956977i \(-0.406290\pi\)
0.290163 + 0.956977i \(0.406290\pi\)
\(332\) −2.02503e8 −0.303702
\(333\) −3.67585e8 −0.545511
\(334\) −2.27717e8 −0.334413
\(335\) 4.83702e8 0.702944
\(336\) −7.43213e7 −0.106887
\(337\) 7.71340e8 1.09785 0.548923 0.835873i \(-0.315038\pi\)
0.548923 + 0.835873i \(0.315038\pi\)
\(338\) −6.86147e7 −0.0966515
\(339\) −7.18283e8 −1.00137
\(340\) −4.14784e8 −0.572329
\(341\) −1.52042e8 −0.207646
\(342\) 3.01316e8 0.407315
\(343\) 1.06083e9 1.41943
\(344\) −1.51904e8 −0.201194
\(345\) −2.40721e8 −0.315607
\(346\) −1.19281e8 −0.154813
\(347\) −1.31263e9 −1.68651 −0.843257 0.537511i \(-0.819365\pi\)
−0.843257 + 0.537511i \(0.819365\pi\)
\(348\) −2.03755e7 −0.0259168
\(349\) −5.67291e8 −0.714360 −0.357180 0.934036i \(-0.616262\pi\)
−0.357180 + 0.934036i \(0.616262\pi\)
\(350\) −5.53026e8 −0.689457
\(351\) −1.44000e8 −0.177741
\(352\) −2.91191e8 −0.355859
\(353\) 1.79304e8 0.216959 0.108480 0.994099i \(-0.465402\pi\)
0.108480 + 0.994099i \(0.465402\pi\)
\(354\) 4.12449e7 0.0494151
\(355\) −1.46091e8 −0.173310
\(356\) 4.47760e8 0.525981
\(357\) 1.37198e9 1.59592
\(358\) −1.74002e8 −0.200430
\(359\) −1.28294e9 −1.46344 −0.731722 0.681603i \(-0.761283\pi\)
−0.731722 + 0.681603i \(0.761283\pi\)
\(360\) −1.86969e8 −0.211208
\(361\) 2.19419e9 2.45470
\(362\) −7.32106e8 −0.811136
\(363\) −4.53630e8 −0.497770
\(364\) 8.13441e8 0.884039
\(365\) 1.00389e9 1.08059
\(366\) 1.46070e8 0.155731
\(367\) 1.71884e7 0.0181512 0.00907560 0.999959i \(-0.497111\pi\)
0.00907560 + 0.999959i \(0.497111\pi\)
\(368\) 9.33583e7 0.0976530
\(369\) 4.63271e8 0.480002
\(370\) −6.44426e8 −0.661404
\(371\) 2.55846e9 2.60118
\(372\) −1.82042e8 −0.183346
\(373\) −8.02127e8 −0.800318 −0.400159 0.916446i \(-0.631045\pi\)
−0.400159 + 0.916446i \(0.631045\pi\)
\(374\) −4.04891e8 −0.400210
\(375\) −5.87921e8 −0.575718
\(376\) 1.08085e8 0.104860
\(377\) −7.59657e7 −0.0730168
\(378\) 2.23973e8 0.213292
\(379\) 9.63664e6 0.00909261 0.00454630 0.999990i \(-0.498553\pi\)
0.00454630 + 0.999990i \(0.498553\pi\)
\(380\) −6.93960e8 −0.648772
\(381\) −5.51725e8 −0.511075
\(382\) 2.39168e8 0.219523
\(383\) 8.26495e8 0.751700 0.375850 0.926681i \(-0.377351\pi\)
0.375850 + 0.926681i \(0.377351\pi\)
\(384\) −3.02395e8 −0.272531
\(385\) −4.30826e8 −0.384760
\(386\) 1.92925e8 0.170739
\(387\) 7.41904e7 0.0650668
\(388\) 1.21870e9 1.05922
\(389\) 7.27469e8 0.626601 0.313300 0.949654i \(-0.398565\pi\)
0.313300 + 0.949654i \(0.398565\pi\)
\(390\) −2.52452e8 −0.215503
\(391\) −1.72341e9 −1.45804
\(392\) −2.26424e9 −1.89854
\(393\) −1.13060e9 −0.939579
\(394\) −1.19509e8 −0.0984383
\(395\) 4.72104e8 0.385432
\(396\) 8.68328e7 0.0702669
\(397\) −1.15679e9 −0.927871 −0.463936 0.885869i \(-0.653563\pi\)
−0.463936 + 0.885869i \(0.653563\pi\)
\(398\) 4.81563e8 0.382880
\(399\) 2.29541e9 1.80907
\(400\) 8.74451e7 0.0683165
\(401\) 9.57807e7 0.0741776 0.0370888 0.999312i \(-0.488192\pi\)
0.0370888 + 0.999312i \(0.488192\pi\)
\(402\) 5.65329e8 0.434020
\(403\) −6.78703e8 −0.516550
\(404\) −5.77019e8 −0.435367
\(405\) 9.13159e7 0.0683052
\(406\) 1.18154e8 0.0876211
\(407\) 8.26394e8 0.607585
\(408\) −1.33858e9 −0.975739
\(409\) −1.12641e9 −0.814079 −0.407040 0.913410i \(-0.633439\pi\)
−0.407040 + 0.913410i \(0.633439\pi\)
\(410\) 8.12176e8 0.581978
\(411\) 1.51392e9 1.07561
\(412\) −1.28457e9 −0.904936
\(413\) 3.14203e8 0.219475
\(414\) −2.81343e8 −0.194866
\(415\) 4.78766e8 0.328817
\(416\) −1.29985e9 −0.885251
\(417\) −1.05413e9 −0.711901
\(418\) −6.77408e8 −0.453663
\(419\) −2.46333e9 −1.63596 −0.817981 0.575245i \(-0.804907\pi\)
−0.817981 + 0.575245i \(0.804907\pi\)
\(420\) −5.15833e8 −0.339732
\(421\) 6.02698e8 0.393652 0.196826 0.980438i \(-0.436937\pi\)
0.196826 + 0.980438i \(0.436937\pi\)
\(422\) −3.01596e8 −0.195358
\(423\) −5.27892e7 −0.0339120
\(424\) −2.49617e9 −1.59035
\(425\) −1.61425e9 −1.02002
\(426\) −1.70745e8 −0.107007
\(427\) 1.11275e9 0.691674
\(428\) 6.66417e8 0.410859
\(429\) 3.23737e8 0.197967
\(430\) 1.30066e8 0.0788902
\(431\) 1.69548e9 1.02005 0.510027 0.860159i \(-0.329636\pi\)
0.510027 + 0.860159i \(0.329636\pi\)
\(432\) −3.54149e7 −0.0211346
\(433\) −5.31537e8 −0.314649 −0.157324 0.987547i \(-0.550287\pi\)
−0.157324 + 0.987547i \(0.550287\pi\)
\(434\) 1.05563e9 0.619867
\(435\) 4.81726e7 0.0280600
\(436\) −9.61000e8 −0.555291
\(437\) −2.88337e9 −1.65278
\(438\) 1.17330e9 0.667191
\(439\) −2.39663e9 −1.35200 −0.675998 0.736904i \(-0.736287\pi\)
−0.675998 + 0.736904i \(0.736287\pi\)
\(440\) 4.20337e8 0.235241
\(441\) 1.10586e9 0.613995
\(442\) −1.80740e9 −0.995580
\(443\) −2.03310e9 −1.11108 −0.555542 0.831489i \(-0.687489\pi\)
−0.555542 + 0.831489i \(0.687489\pi\)
\(444\) 9.89451e8 0.536480
\(445\) −1.05861e9 −0.569478
\(446\) −4.97550e8 −0.265561
\(447\) 1.50800e9 0.798594
\(448\) 2.37408e9 1.24745
\(449\) 3.46417e9 1.80608 0.903040 0.429557i \(-0.141330\pi\)
0.903040 + 0.429557i \(0.141330\pi\)
\(450\) −2.63523e8 −0.136325
\(451\) −1.04151e9 −0.534622
\(452\) 1.93344e9 0.984798
\(453\) −1.83092e9 −0.925393
\(454\) 9.09639e8 0.456219
\(455\) −1.92317e9 −0.957146
\(456\) −2.23953e9 −1.10606
\(457\) −2.70035e8 −0.132347 −0.0661735 0.997808i \(-0.521079\pi\)
−0.0661735 + 0.997808i \(0.521079\pi\)
\(458\) −1.72849e9 −0.840692
\(459\) 6.53766e8 0.315557
\(460\) 6.47962e8 0.310382
\(461\) −1.94305e9 −0.923698 −0.461849 0.886959i \(-0.652814\pi\)
−0.461849 + 0.886959i \(0.652814\pi\)
\(462\) −5.03530e8 −0.237563
\(463\) −1.57266e8 −0.0736378 −0.0368189 0.999322i \(-0.511722\pi\)
−0.0368189 + 0.999322i \(0.511722\pi\)
\(464\) −1.86827e7 −0.00868215
\(465\) 4.30391e8 0.198508
\(466\) 5.72494e8 0.262072
\(467\) −1.00777e9 −0.457881 −0.228941 0.973440i \(-0.573526\pi\)
−0.228941 + 0.973440i \(0.573526\pi\)
\(468\) 3.87614e8 0.174799
\(469\) 4.30666e9 1.92768
\(470\) −9.25465e7 −0.0411166
\(471\) 1.20358e9 0.530764
\(472\) −3.06553e8 −0.134186
\(473\) −1.66793e8 −0.0724708
\(474\) 5.51773e8 0.237978
\(475\) −2.70074e9 −1.15626
\(476\) −3.69305e9 −1.56950
\(477\) 1.21914e9 0.514325
\(478\) −7.94732e7 −0.0332830
\(479\) 1.14824e9 0.477372 0.238686 0.971097i \(-0.423283\pi\)
0.238686 + 0.971097i \(0.423283\pi\)
\(480\) 8.24283e8 0.340198
\(481\) 3.68895e9 1.51145
\(482\) −2.78276e9 −1.13191
\(483\) −2.14326e9 −0.865487
\(484\) 1.22106e9 0.489530
\(485\) −2.88130e9 −1.14681
\(486\) 1.06726e8 0.0421738
\(487\) 2.65456e9 1.04146 0.520729 0.853722i \(-0.325660\pi\)
0.520729 + 0.853722i \(0.325660\pi\)
\(488\) −1.08566e9 −0.422888
\(489\) −2.59923e9 −1.00523
\(490\) 1.93872e9 0.744438
\(491\) −2.25039e9 −0.857971 −0.428986 0.903311i \(-0.641129\pi\)
−0.428986 + 0.903311i \(0.641129\pi\)
\(492\) −1.24701e9 −0.472056
\(493\) 3.44886e8 0.129632
\(494\) −3.02389e9 −1.12855
\(495\) −2.05294e8 −0.0760777
\(496\) −1.66918e8 −0.0614210
\(497\) −1.30073e9 −0.475268
\(498\) 5.59559e8 0.203022
\(499\) 7.47962e8 0.269481 0.134740 0.990881i \(-0.456980\pi\)
0.134740 + 0.990881i \(0.456980\pi\)
\(500\) 1.58254e9 0.566188
\(501\) −8.26624e8 −0.293681
\(502\) 1.95940e9 0.691288
\(503\) 5.29706e9 1.85587 0.927933 0.372746i \(-0.121584\pi\)
0.927933 + 0.372746i \(0.121584\pi\)
\(504\) −1.66468e9 −0.579194
\(505\) 1.36421e9 0.471370
\(506\) 6.32507e8 0.217039
\(507\) −2.49075e8 −0.0848794
\(508\) 1.48511e9 0.502615
\(509\) 2.52094e9 0.847327 0.423663 0.905820i \(-0.360744\pi\)
0.423663 + 0.905820i \(0.360744\pi\)
\(510\) 1.14614e9 0.382597
\(511\) 8.93818e9 2.96330
\(512\) −6.63440e8 −0.218452
\(513\) 1.09379e9 0.357704
\(514\) −2.92167e9 −0.948988
\(515\) 3.03703e9 0.979771
\(516\) −1.99703e8 −0.0639897
\(517\) 1.18679e8 0.0377709
\(518\) −5.73767e9 −1.81377
\(519\) −4.32998e8 −0.135957
\(520\) 1.87635e9 0.585197
\(521\) 6.27028e9 1.94247 0.971236 0.238118i \(-0.0765305\pi\)
0.971236 + 0.238118i \(0.0765305\pi\)
\(522\) 5.63020e7 0.0173251
\(523\) −6.45997e8 −0.197458 −0.0987289 0.995114i \(-0.531478\pi\)
−0.0987289 + 0.995114i \(0.531478\pi\)
\(524\) 3.04329e9 0.924025
\(525\) −2.00751e9 −0.605481
\(526\) −1.79559e9 −0.537969
\(527\) 3.08133e9 0.917067
\(528\) 7.96188e7 0.0235395
\(529\) −7.12574e8 −0.209284
\(530\) 2.13731e9 0.623593
\(531\) 1.49721e8 0.0433963
\(532\) −6.17870e9 −1.77912
\(533\) −4.64922e9 −1.32995
\(534\) −1.23726e9 −0.351614
\(535\) −1.57557e9 −0.444835
\(536\) −4.20181e9 −1.17858
\(537\) −6.31635e8 −0.176018
\(538\) 1.75353e9 0.485484
\(539\) −2.48616e9 −0.683861
\(540\) −2.45800e8 −0.0671745
\(541\) −3.10398e9 −0.842807 −0.421404 0.906873i \(-0.638462\pi\)
−0.421404 + 0.906873i \(0.638462\pi\)
\(542\) −1.11536e9 −0.300898
\(543\) −2.65758e9 −0.712340
\(544\) 5.90135e9 1.57165
\(545\) 2.27204e9 0.601212
\(546\) −2.24771e9 −0.590972
\(547\) −2.73990e9 −0.715779 −0.357890 0.933764i \(-0.616504\pi\)
−0.357890 + 0.933764i \(0.616504\pi\)
\(548\) −4.07509e9 −1.05781
\(549\) 5.30240e8 0.136763
\(550\) 5.92445e8 0.151837
\(551\) 5.77016e8 0.146946
\(552\) 2.09108e9 0.529157
\(553\) 4.20339e9 1.05697
\(554\) −7.08821e8 −0.177114
\(555\) −2.33930e9 −0.580845
\(556\) 2.83747e9 0.700116
\(557\) 2.13952e9 0.524595 0.262297 0.964987i \(-0.415520\pi\)
0.262297 + 0.964987i \(0.415520\pi\)
\(558\) 5.03021e8 0.122565
\(559\) −7.44548e8 −0.180282
\(560\) −4.72978e8 −0.113811
\(561\) −1.46978e9 −0.351464
\(562\) −5.43163e8 −0.129078
\(563\) −2.85723e9 −0.674786 −0.337393 0.941364i \(-0.609545\pi\)
−0.337393 + 0.941364i \(0.609545\pi\)
\(564\) 1.42096e8 0.0333506
\(565\) −4.57113e9 −1.06624
\(566\) 1.33689e9 0.309912
\(567\) 8.13035e8 0.187313
\(568\) 1.26906e9 0.290578
\(569\) −3.13031e9 −0.712352 −0.356176 0.934419i \(-0.615920\pi\)
−0.356176 + 0.934419i \(0.615920\pi\)
\(570\) 1.91756e9 0.433698
\(571\) −4.57416e9 −1.02822 −0.514109 0.857725i \(-0.671877\pi\)
−0.514109 + 0.857725i \(0.671877\pi\)
\(572\) −8.71422e8 −0.194690
\(573\) 8.68191e8 0.192785
\(574\) 7.23124e9 1.59596
\(575\) 2.52173e9 0.553173
\(576\) 1.13128e9 0.246655
\(577\) −5.19496e9 −1.12581 −0.562907 0.826520i \(-0.690317\pi\)
−0.562907 + 0.826520i \(0.690317\pi\)
\(578\) 5.15358e9 1.11010
\(579\) 7.00328e8 0.149943
\(580\) −1.29669e8 −0.0275955
\(581\) 4.26271e9 0.901714
\(582\) −3.36753e9 −0.708078
\(583\) −2.74082e9 −0.572850
\(584\) −8.72056e9 −1.81176
\(585\) −9.16413e8 −0.189254
\(586\) 6.04459e9 1.24087
\(587\) 2.60032e9 0.530632 0.265316 0.964162i \(-0.414524\pi\)
0.265316 + 0.964162i \(0.414524\pi\)
\(588\) −2.97671e9 −0.603831
\(589\) 5.15526e9 1.03955
\(590\) 2.62481e8 0.0526158
\(591\) −4.33824e8 −0.0864485
\(592\) 9.07248e8 0.179721
\(593\) 7.30947e9 1.43944 0.719721 0.694263i \(-0.244269\pi\)
0.719721 + 0.694263i \(0.244269\pi\)
\(594\) −2.39938e8 −0.0469728
\(595\) 8.73125e9 1.69929
\(596\) −4.05918e9 −0.785374
\(597\) 1.74810e9 0.336245
\(598\) 2.82346e9 0.539917
\(599\) 1.02859e9 0.195545 0.0977725 0.995209i \(-0.468828\pi\)
0.0977725 + 0.995209i \(0.468828\pi\)
\(600\) 1.95864e9 0.370190
\(601\) −2.84013e9 −0.533676 −0.266838 0.963741i \(-0.585979\pi\)
−0.266838 + 0.963741i \(0.585979\pi\)
\(602\) 1.15804e9 0.216340
\(603\) 2.05217e9 0.381156
\(604\) 4.92840e9 0.910074
\(605\) −2.88689e9 −0.530013
\(606\) 1.59443e9 0.291039
\(607\) 8.39067e9 1.52278 0.761388 0.648296i \(-0.224518\pi\)
0.761388 + 0.648296i \(0.224518\pi\)
\(608\) 9.87333e9 1.78156
\(609\) 4.28907e8 0.0769489
\(610\) 9.29582e8 0.165819
\(611\) 5.29773e8 0.0939605
\(612\) −1.75978e9 −0.310333
\(613\) −2.69978e9 −0.473388 −0.236694 0.971584i \(-0.576064\pi\)
−0.236694 + 0.971584i \(0.576064\pi\)
\(614\) −1.44033e9 −0.251115
\(615\) 2.94824e9 0.511093
\(616\) 3.74249e9 0.645101
\(617\) 3.54215e9 0.607112 0.303556 0.952814i \(-0.401826\pi\)
0.303556 + 0.952814i \(0.401826\pi\)
\(618\) 3.54955e9 0.604941
\(619\) −7.53401e9 −1.27676 −0.638380 0.769721i \(-0.720395\pi\)
−0.638380 + 0.769721i \(0.720395\pi\)
\(620\) −1.15851e9 −0.195222
\(621\) −1.02129e9 −0.171131
\(622\) 1.31475e9 0.219067
\(623\) −9.42539e9 −1.56168
\(624\) 3.55411e8 0.0585579
\(625\) 5.54007e7 0.00907685
\(626\) 1.01554e9 0.165457
\(627\) −2.45903e9 −0.398407
\(628\) −3.23974e9 −0.521978
\(629\) −1.67479e10 −2.68339
\(630\) 1.42536e9 0.227108
\(631\) 1.21541e10 1.92584 0.962920 0.269787i \(-0.0869533\pi\)
0.962920 + 0.269787i \(0.0869533\pi\)
\(632\) −4.10106e9 −0.646228
\(633\) −1.09481e9 −0.171564
\(634\) −1.24271e9 −0.193668
\(635\) −3.51116e9 −0.544179
\(636\) −3.28162e9 −0.505811
\(637\) −1.10980e10 −1.70120
\(638\) −1.26576e8 −0.0192966
\(639\) −6.19812e8 −0.0939738
\(640\) −1.92443e9 −0.290183
\(641\) 6.74739e9 1.01189 0.505944 0.862566i \(-0.331144\pi\)
0.505944 + 0.862566i \(0.331144\pi\)
\(642\) −1.84145e9 −0.274655
\(643\) −8.85998e9 −1.31430 −0.657150 0.753760i \(-0.728238\pi\)
−0.657150 + 0.753760i \(0.728238\pi\)
\(644\) 5.76915e9 0.851160
\(645\) 4.72145e8 0.0692814
\(646\) 1.37286e10 2.00360
\(647\) −5.90409e9 −0.857015 −0.428507 0.903538i \(-0.640960\pi\)
−0.428507 + 0.903538i \(0.640960\pi\)
\(648\) −7.93240e8 −0.114523
\(649\) −3.36599e8 −0.0483344
\(650\) 2.64462e9 0.377717
\(651\) 3.83200e9 0.544367
\(652\) 6.99651e9 0.988587
\(653\) −8.95278e9 −1.25823 −0.629117 0.777310i \(-0.716584\pi\)
−0.629117 + 0.777310i \(0.716584\pi\)
\(654\) 2.65545e9 0.371207
\(655\) −7.19507e9 −1.00044
\(656\) −1.14341e9 −0.158139
\(657\) 4.25914e9 0.585927
\(658\) −8.23991e8 −0.112754
\(659\) 1.19309e10 1.62396 0.811979 0.583687i \(-0.198391\pi\)
0.811979 + 0.583687i \(0.198391\pi\)
\(660\) 5.52601e8 0.0748183
\(661\) −3.52536e9 −0.474786 −0.237393 0.971414i \(-0.576293\pi\)
−0.237393 + 0.971414i \(0.576293\pi\)
\(662\) 2.84788e9 0.381521
\(663\) −6.56095e9 −0.874318
\(664\) −4.15892e9 −0.551306
\(665\) 1.46079e10 1.92625
\(666\) −2.73407e9 −0.358632
\(667\) −5.38769e8 −0.0703011
\(668\) 2.22507e9 0.288820
\(669\) −1.80613e9 −0.233216
\(670\) 3.59773e9 0.462133
\(671\) −1.19207e9 −0.152326
\(672\) 7.33903e9 0.932924
\(673\) 2.13472e9 0.269953 0.134976 0.990849i \(-0.456904\pi\)
0.134976 + 0.990849i \(0.456904\pi\)
\(674\) 5.73716e9 0.721751
\(675\) −9.56603e8 −0.119720
\(676\) 6.70450e8 0.0834743
\(677\) 1.03034e10 1.27621 0.638103 0.769951i \(-0.279719\pi\)
0.638103 + 0.769951i \(0.279719\pi\)
\(678\) −5.34253e9 −0.658329
\(679\) −2.56537e10 −3.14490
\(680\) −8.51867e9 −1.03894
\(681\) 3.30204e9 0.400652
\(682\) −1.13088e9 −0.136512
\(683\) 4.34566e9 0.521895 0.260947 0.965353i \(-0.415965\pi\)
0.260947 + 0.965353i \(0.415965\pi\)
\(684\) −2.94422e9 −0.351782
\(685\) 9.63450e9 1.14528
\(686\) 7.89033e9 0.933170
\(687\) −6.27449e9 −0.738295
\(688\) −1.83112e8 −0.0214366
\(689\) −1.22348e10 −1.42505
\(690\) −1.79046e9 −0.207488
\(691\) 1.26968e9 0.146393 0.0731965 0.997318i \(-0.476680\pi\)
0.0731965 + 0.997318i \(0.476680\pi\)
\(692\) 1.16553e9 0.133706
\(693\) −1.82784e9 −0.208628
\(694\) −9.76324e9 −1.10876
\(695\) −6.70847e9 −0.758013
\(696\) −4.18464e8 −0.0470464
\(697\) 2.11076e10 2.36115
\(698\) −4.21946e9 −0.469638
\(699\) 2.07818e9 0.230151
\(700\) 5.40374e9 0.595458
\(701\) −1.71178e9 −0.187687 −0.0938435 0.995587i \(-0.529915\pi\)
−0.0938435 + 0.995587i \(0.529915\pi\)
\(702\) −1.07106e9 −0.116852
\(703\) −2.80204e10 −3.04180
\(704\) −2.54330e9 −0.274722
\(705\) −3.35948e8 −0.0361086
\(706\) 1.33365e9 0.142634
\(707\) 1.21463e10 1.29264
\(708\) −4.03014e8 −0.0426779
\(709\) 1.22235e10 1.28805 0.644025 0.765004i \(-0.277263\pi\)
0.644025 + 0.765004i \(0.277263\pi\)
\(710\) −1.08661e9 −0.113938
\(711\) 2.00297e9 0.208992
\(712\) 9.19592e9 0.954806
\(713\) −4.81355e9 −0.497338
\(714\) 1.02047e10 1.04919
\(715\) 2.06025e9 0.210790
\(716\) 1.70021e9 0.173104
\(717\) −2.88492e8 −0.0292292
\(718\) −9.54240e9 −0.962104
\(719\) −5.76282e8 −0.0578207 −0.0289104 0.999582i \(-0.509204\pi\)
−0.0289104 + 0.999582i \(0.509204\pi\)
\(720\) −2.25379e8 −0.0225035
\(721\) 2.70403e10 2.68682
\(722\) 1.63202e10 1.61378
\(723\) −1.01016e10 −0.994043
\(724\) 7.15357e9 0.700548
\(725\) −5.04644e8 −0.0491816
\(726\) −3.37406e9 −0.327246
\(727\) 3.68604e9 0.355787 0.177894 0.984050i \(-0.443072\pi\)
0.177894 + 0.984050i \(0.443072\pi\)
\(728\) 1.67061e10 1.60478
\(729\) 3.87420e8 0.0370370
\(730\) 7.46685e9 0.710407
\(731\) 3.38027e9 0.320067
\(732\) −1.42728e9 −0.134499
\(733\) −5.75370e9 −0.539614 −0.269807 0.962914i \(-0.586960\pi\)
−0.269807 + 0.962914i \(0.586960\pi\)
\(734\) 1.27846e8 0.0119331
\(735\) 7.03765e9 0.653765
\(736\) −9.21889e9 −0.852327
\(737\) −4.61363e9 −0.424528
\(738\) 3.44577e9 0.315565
\(739\) −7.57514e9 −0.690455 −0.345227 0.938519i \(-0.612198\pi\)
−0.345227 + 0.938519i \(0.612198\pi\)
\(740\) 6.29683e9 0.571230
\(741\) −1.09769e10 −0.991095
\(742\) 1.90296e10 1.71008
\(743\) 9.30976e9 0.832679 0.416340 0.909209i \(-0.363313\pi\)
0.416340 + 0.909209i \(0.363313\pi\)
\(744\) −3.73870e9 −0.332825
\(745\) 9.59688e9 0.850321
\(746\) −5.96615e9 −0.526149
\(747\) 2.03123e9 0.178294
\(748\) 3.95628e9 0.345646
\(749\) −1.40281e10 −1.21987
\(750\) −4.37291e9 −0.378491
\(751\) 1.04130e10 0.897091 0.448546 0.893760i \(-0.351942\pi\)
0.448546 + 0.893760i \(0.351942\pi\)
\(752\) 1.30290e8 0.0111725
\(753\) 7.11271e9 0.607089
\(754\) −5.65026e8 −0.0480030
\(755\) −1.16519e10 −0.985334
\(756\) −2.18849e9 −0.184212
\(757\) −5.37115e9 −0.450020 −0.225010 0.974356i \(-0.572241\pi\)
−0.225010 + 0.974356i \(0.572241\pi\)
\(758\) 7.16765e7 0.00597771
\(759\) 2.29603e9 0.190604
\(760\) −1.42523e10 −1.17771
\(761\) −2.00035e10 −1.64535 −0.822677 0.568509i \(-0.807521\pi\)
−0.822677 + 0.568509i \(0.807521\pi\)
\(762\) −4.10368e9 −0.335994
\(763\) 2.02292e10 1.64870
\(764\) −2.33696e9 −0.189594
\(765\) 4.16054e9 0.335997
\(766\) 6.14740e9 0.494186
\(767\) −1.50255e9 −0.120239
\(768\) −7.61228e9 −0.606388
\(769\) −1.99672e10 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(770\) −3.20445e9 −0.252950
\(771\) −1.06058e10 −0.833401
\(772\) −1.88511e9 −0.147461
\(773\) −1.32942e10 −1.03522 −0.517611 0.855616i \(-0.673179\pi\)
−0.517611 + 0.855616i \(0.673179\pi\)
\(774\) 5.51822e8 0.0427765
\(775\) −4.50867e9 −0.347930
\(776\) 2.50292e10 1.92278
\(777\) −2.08280e10 −1.59285
\(778\) 5.41085e9 0.411943
\(779\) 3.53143e10 2.67652
\(780\) 2.46676e9 0.186121
\(781\) 1.39344e9 0.104667
\(782\) −1.28186e10 −0.958552
\(783\) 2.04379e8 0.0152149
\(784\) −2.72940e9 −0.202284
\(785\) 7.65954e9 0.565143
\(786\) −8.40927e9 −0.617702
\(787\) −1.67731e10 −1.22659 −0.613297 0.789853i \(-0.710157\pi\)
−0.613297 + 0.789853i \(0.710157\pi\)
\(788\) 1.16775e9 0.0850174
\(789\) −6.51808e9 −0.472444
\(790\) 3.51147e9 0.253393
\(791\) −4.06992e10 −2.92394
\(792\) 1.78334e9 0.127554
\(793\) −5.32130e9 −0.378932
\(794\) −8.60410e9 −0.610005
\(795\) 7.75854e9 0.547640
\(796\) −4.70546e9 −0.330679
\(797\) −1.36721e10 −0.956603 −0.478301 0.878196i \(-0.658747\pi\)
−0.478301 + 0.878196i \(0.658747\pi\)
\(798\) 1.70731e10 1.18933
\(799\) −2.40518e9 −0.166815
\(800\) −8.63498e9 −0.596275
\(801\) −4.49131e9 −0.308787
\(802\) 7.12408e8 0.0487662
\(803\) −9.57528e9 −0.652600
\(804\) −5.52395e9 −0.374847
\(805\) −1.36397e10 −0.921548
\(806\) −5.04813e9 −0.339593
\(807\) 6.36540e9 0.426352
\(808\) −1.18506e10 −0.790314
\(809\) 2.47296e10 1.64209 0.821046 0.570862i \(-0.193391\pi\)
0.821046 + 0.570862i \(0.193391\pi\)
\(810\) 6.79200e8 0.0449055
\(811\) −5.97641e9 −0.393430 −0.196715 0.980461i \(-0.563027\pi\)
−0.196715 + 0.980461i \(0.563027\pi\)
\(812\) −1.15451e9 −0.0756751
\(813\) −4.04883e9 −0.264248
\(814\) 6.14664e9 0.399441
\(815\) −1.65414e10 −1.07034
\(816\) −1.61358e9 −0.103962
\(817\) 5.65540e9 0.362816
\(818\) −8.37817e9 −0.535196
\(819\) −8.15932e9 −0.518991
\(820\) −7.93596e9 −0.502633
\(821\) 6.86651e9 0.433047 0.216523 0.976277i \(-0.430528\pi\)
0.216523 + 0.976277i \(0.430528\pi\)
\(822\) 1.12604e10 0.707133
\(823\) −1.06812e10 −0.667916 −0.333958 0.942588i \(-0.608384\pi\)
−0.333958 + 0.942588i \(0.608384\pi\)
\(824\) −2.63820e10 −1.64272
\(825\) 2.15061e9 0.133344
\(826\) 2.33701e9 0.144288
\(827\) 1.28871e10 0.792291 0.396146 0.918188i \(-0.370348\pi\)
0.396146 + 0.918188i \(0.370348\pi\)
\(828\) 2.74907e9 0.168298
\(829\) 2.54525e10 1.55164 0.775819 0.630956i \(-0.217337\pi\)
0.775819 + 0.630956i \(0.217337\pi\)
\(830\) 3.56102e9 0.216173
\(831\) −2.57306e9 −0.155541
\(832\) −1.13531e10 −0.683411
\(833\) 5.03852e10 3.02027
\(834\) −7.84056e9 −0.468021
\(835\) −5.26061e9 −0.312704
\(836\) 6.61911e9 0.391812
\(837\) 1.82599e9 0.107637
\(838\) −1.83220e10 −1.07552
\(839\) 1.16543e10 0.681269 0.340635 0.940196i \(-0.389358\pi\)
0.340635 + 0.940196i \(0.389358\pi\)
\(840\) −1.05940e10 −0.616711
\(841\) −1.71421e10 −0.993750
\(842\) 4.48282e9 0.258797
\(843\) −1.97171e9 −0.113356
\(844\) 2.94696e9 0.168724
\(845\) −1.58510e9 −0.0903773
\(846\) −3.92641e8 −0.0222946
\(847\) −2.57035e10 −1.45345
\(848\) −3.00898e9 −0.169447
\(849\) 4.85299e9 0.272165
\(850\) −1.20067e10 −0.670588
\(851\) 2.61631e10 1.45524
\(852\) 1.66838e9 0.0924181
\(853\) −8.84397e9 −0.487894 −0.243947 0.969789i \(-0.578442\pi\)
−0.243947 + 0.969789i \(0.578442\pi\)
\(854\) 8.27657e9 0.454724
\(855\) 6.96085e9 0.380873
\(856\) 1.36866e10 0.745826
\(857\) −9.53214e9 −0.517318 −0.258659 0.965969i \(-0.583281\pi\)
−0.258659 + 0.965969i \(0.583281\pi\)
\(858\) 2.40793e9 0.130148
\(859\) −1.01239e10 −0.544971 −0.272485 0.962160i \(-0.587846\pi\)
−0.272485 + 0.962160i \(0.587846\pi\)
\(860\) −1.27090e9 −0.0681345
\(861\) 2.62498e10 1.40157
\(862\) 1.26108e10 0.670608
\(863\) 3.34111e9 0.176951 0.0884755 0.996078i \(-0.471801\pi\)
0.0884755 + 0.996078i \(0.471801\pi\)
\(864\) 3.49713e9 0.184465
\(865\) −2.75558e9 −0.144763
\(866\) −3.95353e9 −0.206858
\(867\) 1.87078e10 0.974888
\(868\) −1.03148e10 −0.535356
\(869\) −4.50301e9 −0.232773
\(870\) 3.58304e8 0.0184473
\(871\) −2.05948e10 −1.05608
\(872\) −1.97366e10 −1.00801
\(873\) −1.22243e10 −0.621834
\(874\) −2.14463e10 −1.08658
\(875\) −3.33127e10 −1.68105
\(876\) −1.14646e10 −0.576228
\(877\) −1.70327e10 −0.852679 −0.426339 0.904563i \(-0.640197\pi\)
−0.426339 + 0.904563i \(0.640197\pi\)
\(878\) −1.78259e10 −0.888835
\(879\) 2.19422e10 1.08973
\(880\) 5.06691e8 0.0250642
\(881\) −6.92335e9 −0.341115 −0.170558 0.985348i \(-0.554557\pi\)
−0.170558 + 0.985348i \(0.554557\pi\)
\(882\) 8.22528e9 0.403655
\(883\) −2.45719e9 −0.120109 −0.0600547 0.998195i \(-0.519128\pi\)
−0.0600547 + 0.998195i \(0.519128\pi\)
\(884\) 1.76605e10 0.859845
\(885\) 9.52821e8 0.0462072
\(886\) −1.51220e10 −0.730453
\(887\) 1.99592e10 0.960307 0.480154 0.877184i \(-0.340581\pi\)
0.480154 + 0.877184i \(0.340581\pi\)
\(888\) 2.03209e10 0.973865
\(889\) −3.12617e10 −1.49230
\(890\) −7.87387e9 −0.374389
\(891\) −8.70987e8 −0.0412515
\(892\) 4.86167e9 0.229355
\(893\) −4.02402e9 −0.189095
\(894\) 1.12164e10 0.525015
\(895\) −4.01971e9 −0.187419
\(896\) −1.71342e10 −0.795768
\(897\) 1.02493e10 0.474155
\(898\) 2.57662e10 1.18736
\(899\) 9.63280e8 0.0442174
\(900\) 2.57494e9 0.117739
\(901\) 5.55463e10 2.52999
\(902\) −7.74667e9 −0.351473
\(903\) 4.20376e9 0.189990
\(904\) 3.97083e10 1.78769
\(905\) −1.69128e10 −0.758481
\(906\) −1.36182e10 −0.608376
\(907\) 1.22070e10 0.543227 0.271614 0.962406i \(-0.412443\pi\)
0.271614 + 0.962406i \(0.412443\pi\)
\(908\) −8.88829e9 −0.394019
\(909\) 5.78785e9 0.255590
\(910\) −1.43044e10 −0.629251
\(911\) 1.78824e10 0.783629 0.391815 0.920044i \(-0.371847\pi\)
0.391815 + 0.920044i \(0.371847\pi\)
\(912\) −2.69962e9 −0.117847
\(913\) −4.56655e9 −0.198582
\(914\) −2.00850e9 −0.0870081
\(915\) 3.37443e9 0.145622
\(916\) 1.68894e10 0.726074
\(917\) −6.40616e10 −2.74350
\(918\) 4.86265e9 0.207455
\(919\) −1.01138e10 −0.429843 −0.214921 0.976631i \(-0.568949\pi\)
−0.214921 + 0.976631i \(0.568949\pi\)
\(920\) 1.33076e10 0.563432
\(921\) −5.22847e9 −0.220529
\(922\) −1.44522e10 −0.607262
\(923\) 6.22020e9 0.260375
\(924\) 4.92010e9 0.205174
\(925\) 2.45059e10 1.01806
\(926\) −1.16973e9 −0.0484113
\(927\) 1.28850e10 0.531259
\(928\) 1.84487e9 0.0757788
\(929\) 7.28094e9 0.297942 0.148971 0.988842i \(-0.452404\pi\)
0.148971 + 0.988842i \(0.452404\pi\)
\(930\) 3.20121e9 0.130504
\(931\) 8.42976e10 3.42367
\(932\) −5.59397e9 −0.226342
\(933\) 4.77262e9 0.192385
\(934\) −7.49571e9 −0.301022
\(935\) −9.35360e9 −0.374230
\(936\) 7.96067e9 0.317310
\(937\) −2.64374e10 −1.04986 −0.524928 0.851147i \(-0.675908\pi\)
−0.524928 + 0.851147i \(0.675908\pi\)
\(938\) 3.20325e10 1.26731
\(939\) 3.68645e9 0.145304
\(940\) 9.04292e8 0.0355109
\(941\) −7.30500e9 −0.285796 −0.142898 0.989737i \(-0.545642\pi\)
−0.142898 + 0.989737i \(0.545642\pi\)
\(942\) 8.95211e9 0.348937
\(943\) −3.29735e10 −1.28049
\(944\) −3.69531e8 −0.0142971
\(945\) 5.17412e9 0.199446
\(946\) −1.24059e9 −0.0476441
\(947\) 6.35717e9 0.243242 0.121621 0.992577i \(-0.461191\pi\)
0.121621 + 0.992577i \(0.461191\pi\)
\(948\) −5.39150e9 −0.205533
\(949\) −4.27432e10 −1.62344
\(950\) −2.00879e10 −0.760154
\(951\) −4.51111e9 −0.170079
\(952\) −7.58463e10 −2.84908
\(953\) −6.22176e9 −0.232856 −0.116428 0.993199i \(-0.537144\pi\)
−0.116428 + 0.993199i \(0.537144\pi\)
\(954\) 9.06782e9 0.338130
\(955\) 5.52514e9 0.205273
\(956\) 7.76550e8 0.0287453
\(957\) −4.59479e8 −0.0169462
\(958\) 8.54047e9 0.313836
\(959\) 8.57811e10 3.14070
\(960\) 7.19941e9 0.262632
\(961\) −1.89063e10 −0.687188
\(962\) 2.74381e10 0.993668
\(963\) −6.68457e9 −0.241202
\(964\) 2.71910e10 0.977587
\(965\) 4.45686e9 0.159655
\(966\) −1.59414e10 −0.568993
\(967\) −3.37902e10 −1.20171 −0.600853 0.799360i \(-0.705172\pi\)
−0.600853 + 0.799360i \(0.705172\pi\)
\(968\) 2.50777e10 0.888637
\(969\) 4.98354e10 1.75956
\(970\) −2.14308e10 −0.753943
\(971\) −4.96021e10 −1.73873 −0.869366 0.494169i \(-0.835472\pi\)
−0.869366 + 0.494169i \(0.835472\pi\)
\(972\) −1.04284e9 −0.0364239
\(973\) −5.97291e10 −2.07870
\(974\) 1.97444e10 0.684680
\(975\) 9.60012e9 0.331711
\(976\) −1.30870e9 −0.0450574
\(977\) −3.46137e10 −1.18745 −0.593727 0.804667i \(-0.702344\pi\)
−0.593727 + 0.804667i \(0.702344\pi\)
\(978\) −1.93329e10 −0.660861
\(979\) 1.00972e10 0.343924
\(980\) −1.89437e10 −0.642943
\(981\) 9.63943e9 0.325994
\(982\) −1.67382e10 −0.564052
\(983\) −9.52456e9 −0.319822 −0.159911 0.987131i \(-0.551121\pi\)
−0.159911 + 0.987131i \(0.551121\pi\)
\(984\) −2.56107e10 −0.856916
\(985\) −2.76084e9 −0.0920480
\(986\) 2.56523e9 0.0852231
\(987\) −2.99113e9 −0.0990205
\(988\) 2.95471e10 0.974689
\(989\) −5.28054e9 −0.173577
\(990\) −1.52696e9 −0.0500154
\(991\) 4.31860e10 1.40956 0.704782 0.709424i \(-0.251045\pi\)
0.704782 + 0.709424i \(0.251045\pi\)
\(992\) 1.64827e10 0.536090
\(993\) 1.03379e10 0.335052
\(994\) −9.67469e9 −0.312453
\(995\) 1.11248e10 0.358025
\(996\) −5.46758e9 −0.175343
\(997\) 5.74177e9 0.183490 0.0917451 0.995783i \(-0.470755\pi\)
0.0917451 + 0.995783i \(0.470755\pi\)
\(998\) 5.56327e9 0.177163
\(999\) −9.92480e9 −0.314951
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.1.11 18
3.2 odd 2 531.8.a.e.1.8 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.11 18 1.1 even 1 trivial
531.8.a.e.1.8 18 3.2 odd 2