Properties

Label 177.8.a.c.1.17
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $17$
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: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Root \(21.8139\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+21.8139 q^{2} -27.0000 q^{3} +347.844 q^{4} +424.937 q^{5} -588.974 q^{6} +1192.89 q^{7} +4795.65 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+21.8139 q^{2} -27.0000 q^{3} +347.844 q^{4} +424.937 q^{5} -588.974 q^{6} +1192.89 q^{7} +4795.65 q^{8} +729.000 q^{9} +9269.52 q^{10} -3093.76 q^{11} -9391.79 q^{12} +1043.61 q^{13} +26021.4 q^{14} -11473.3 q^{15} +60087.5 q^{16} -16626.4 q^{17} +15902.3 q^{18} +25587.3 q^{19} +147812. q^{20} -32207.9 q^{21} -67486.8 q^{22} -98700.2 q^{23} -129482. q^{24} +102447. q^{25} +22765.2 q^{26} -19683.0 q^{27} +414938. q^{28} -241457. q^{29} -250277. q^{30} -117145. q^{31} +696897. q^{32} +83531.5 q^{33} -362686. q^{34} +506902. q^{35} +253578. q^{36} +73455.9 q^{37} +558158. q^{38} -28177.5 q^{39} +2.03785e6 q^{40} -333204. q^{41} -702579. q^{42} +608971. q^{43} -1.07615e6 q^{44} +309779. q^{45} -2.15303e6 q^{46} -923874. q^{47} -1.62236e6 q^{48} +599434. q^{49} +2.23475e6 q^{50} +448913. q^{51} +363014. q^{52} +617858. q^{53} -429362. q^{54} -1.31465e6 q^{55} +5.72066e6 q^{56} -690858. q^{57} -5.26710e6 q^{58} -205379. q^{59} -3.99092e6 q^{60} +3.02318e6 q^{61} -2.55538e6 q^{62} +869614. q^{63} +7.51081e6 q^{64} +443469. q^{65} +1.82214e6 q^{66} +4.24635e6 q^{67} -5.78339e6 q^{68} +2.66490e6 q^{69} +1.10575e7 q^{70} +809917. q^{71} +3.49603e6 q^{72} +1.31862e6 q^{73} +1.60236e6 q^{74} -2.76606e6 q^{75} +8.90040e6 q^{76} -3.69050e6 q^{77} -614660. q^{78} +5.69982e6 q^{79} +2.55334e7 q^{80} +531441. q^{81} -7.26847e6 q^{82} -3.55052e6 q^{83} -1.12033e7 q^{84} -7.06517e6 q^{85} +1.32840e7 q^{86} +6.51933e6 q^{87} -1.48366e7 q^{88} -1.13503e7 q^{89} +6.75748e6 q^{90} +1.24491e6 q^{91} -3.43323e7 q^{92} +3.16291e6 q^{93} -2.01532e7 q^{94} +1.08730e7 q^{95} -1.88162e7 q^{96} -3.73640e6 q^{97} +1.30760e7 q^{98} -2.25535e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.8139 1.92809 0.964045 0.265738i \(-0.0856157\pi\)
0.964045 + 0.265738i \(0.0856157\pi\)
\(3\) −27.0000 −0.577350
\(4\) 347.844 2.71753
\(5\) 424.937 1.52030 0.760151 0.649747i \(-0.225125\pi\)
0.760151 + 0.649747i \(0.225125\pi\)
\(6\) −588.974 −1.11318
\(7\) 1192.89 1.31449 0.657243 0.753679i \(-0.271723\pi\)
0.657243 + 0.753679i \(0.271723\pi\)
\(8\) 4795.65 3.31156
\(9\) 729.000 0.333333
\(10\) 9269.52 2.93128
\(11\) −3093.76 −0.700829 −0.350415 0.936595i \(-0.613959\pi\)
−0.350415 + 0.936595i \(0.613959\pi\)
\(12\) −9391.79 −1.56897
\(13\) 1043.61 0.131746 0.0658729 0.997828i \(-0.479017\pi\)
0.0658729 + 0.997828i \(0.479017\pi\)
\(14\) 26021.4 2.53445
\(15\) −11473.3 −0.877746
\(16\) 60087.5 3.66745
\(17\) −16626.4 −0.820780 −0.410390 0.911910i \(-0.634607\pi\)
−0.410390 + 0.911910i \(0.634607\pi\)
\(18\) 15902.3 0.642697
\(19\) 25587.3 0.855829 0.427915 0.903819i \(-0.359248\pi\)
0.427915 + 0.903819i \(0.359248\pi\)
\(20\) 147812. 4.13147
\(21\) −32207.9 −0.758919
\(22\) −67486.8 −1.35126
\(23\) −98700.2 −1.69149 −0.845747 0.533584i \(-0.820845\pi\)
−0.845747 + 0.533584i \(0.820845\pi\)
\(24\) −129482. −1.91193
\(25\) 102447. 1.31132
\(26\) 22765.2 0.254018
\(27\) −19683.0 −0.192450
\(28\) 414938. 3.57216
\(29\) −241457. −1.83842 −0.919212 0.393762i \(-0.871173\pi\)
−0.919212 + 0.393762i \(0.871173\pi\)
\(30\) −250277. −1.69237
\(31\) −117145. −0.706248 −0.353124 0.935576i \(-0.614881\pi\)
−0.353124 + 0.935576i \(0.614881\pi\)
\(32\) 696897. 3.75962
\(33\) 83531.5 0.404624
\(34\) −362686. −1.58254
\(35\) 506902. 1.99841
\(36\) 253578. 0.905844
\(37\) 73455.9 0.238408 0.119204 0.992870i \(-0.461966\pi\)
0.119204 + 0.992870i \(0.461966\pi\)
\(38\) 558158. 1.65012
\(39\) −28177.5 −0.0760635
\(40\) 2.03785e6 5.03457
\(41\) −333204. −0.755035 −0.377517 0.926003i \(-0.623222\pi\)
−0.377517 + 0.926003i \(0.623222\pi\)
\(42\) −702579. −1.46326
\(43\) 608971. 1.16804 0.584019 0.811740i \(-0.301479\pi\)
0.584019 + 0.811740i \(0.301479\pi\)
\(44\) −1.07615e6 −1.90453
\(45\) 309779. 0.506767
\(46\) −2.15303e6 −3.26135
\(47\) −923874. −1.29799 −0.648993 0.760794i \(-0.724810\pi\)
−0.648993 + 0.760794i \(0.724810\pi\)
\(48\) −1.62236e6 −2.11740
\(49\) 599434. 0.727872
\(50\) 2.23475e6 2.52834
\(51\) 448913. 0.473878
\(52\) 363014. 0.358024
\(53\) 617858. 0.570064 0.285032 0.958518i \(-0.407996\pi\)
0.285032 + 0.958518i \(0.407996\pi\)
\(54\) −429362. −0.371061
\(55\) −1.31465e6 −1.06547
\(56\) 5.72066e6 4.35299
\(57\) −690858. −0.494113
\(58\) −5.26710e6 −3.54465
\(59\) −205379. −0.130189
\(60\) −3.99092e6 −2.38530
\(61\) 3.02318e6 1.70533 0.852667 0.522455i \(-0.174984\pi\)
0.852667 + 0.522455i \(0.174984\pi\)
\(62\) −2.55538e6 −1.36171
\(63\) 869614. 0.438162
\(64\) 7.51081e6 3.58143
\(65\) 443469. 0.200293
\(66\) 1.82214e6 0.780152
\(67\) 4.24635e6 1.72486 0.862429 0.506177i \(-0.168942\pi\)
0.862429 + 0.506177i \(0.168942\pi\)
\(68\) −5.78339e6 −2.23050
\(69\) 2.66490e6 0.976584
\(70\) 1.10575e7 3.85312
\(71\) 809917. 0.268557 0.134278 0.990944i \(-0.457128\pi\)
0.134278 + 0.990944i \(0.457128\pi\)
\(72\) 3.49603e6 1.10385
\(73\) 1.31862e6 0.396726 0.198363 0.980129i \(-0.436437\pi\)
0.198363 + 0.980129i \(0.436437\pi\)
\(74\) 1.60236e6 0.459672
\(75\) −2.76606e6 −0.757089
\(76\) 8.90040e6 2.32574
\(77\) −3.69050e6 −0.921230
\(78\) −614660. −0.146657
\(79\) 5.69982e6 1.30067 0.650334 0.759649i \(-0.274629\pi\)
0.650334 + 0.759649i \(0.274629\pi\)
\(80\) 2.55334e7 5.57563
\(81\) 531441. 0.111111
\(82\) −7.26847e6 −1.45577
\(83\) −3.55052e6 −0.681583 −0.340791 0.940139i \(-0.610695\pi\)
−0.340791 + 0.940139i \(0.610695\pi\)
\(84\) −1.12033e7 −2.06239
\(85\) −7.06517e6 −1.24783
\(86\) 1.32840e7 2.25208
\(87\) 6.51933e6 1.06142
\(88\) −1.48366e7 −2.32084
\(89\) −1.13503e7 −1.70664 −0.853318 0.521390i \(-0.825414\pi\)
−0.853318 + 0.521390i \(0.825414\pi\)
\(90\) 6.75748e6 0.977093
\(91\) 1.24491e6 0.173178
\(92\) −3.43323e7 −4.59669
\(93\) 3.16291e6 0.407753
\(94\) −2.01532e7 −2.50264
\(95\) 1.08730e7 1.30112
\(96\) −1.88162e7 −2.17062
\(97\) −3.73640e6 −0.415673 −0.207837 0.978164i \(-0.566642\pi\)
−0.207837 + 0.978164i \(0.566642\pi\)
\(98\) 1.30760e7 1.40340
\(99\) −2.25535e6 −0.233610
\(100\) 3.56354e7 3.56354
\(101\) 1.81426e6 0.175216 0.0876080 0.996155i \(-0.472078\pi\)
0.0876080 + 0.996155i \(0.472078\pi\)
\(102\) 9.79251e6 0.913679
\(103\) −2.08845e7 −1.88318 −0.941592 0.336755i \(-0.890671\pi\)
−0.941592 + 0.336755i \(0.890671\pi\)
\(104\) 5.00479e6 0.436284
\(105\) −1.36863e7 −1.15378
\(106\) 1.34779e7 1.09913
\(107\) 3.03824e6 0.239761 0.119881 0.992788i \(-0.461749\pi\)
0.119881 + 0.992788i \(0.461749\pi\)
\(108\) −6.84662e6 −0.522989
\(109\) −7.65394e6 −0.566099 −0.283049 0.959105i \(-0.591346\pi\)
−0.283049 + 0.959105i \(0.591346\pi\)
\(110\) −2.86777e7 −2.05433
\(111\) −1.98331e6 −0.137645
\(112\) 7.16775e7 4.82081
\(113\) 1.47494e6 0.0961611 0.0480805 0.998843i \(-0.484690\pi\)
0.0480805 + 0.998843i \(0.484690\pi\)
\(114\) −1.50703e7 −0.952695
\(115\) −4.19414e7 −2.57158
\(116\) −8.39892e7 −4.99598
\(117\) 760793. 0.0439153
\(118\) −4.48011e6 −0.251016
\(119\) −1.98334e7 −1.07890
\(120\) −5.50219e7 −2.90671
\(121\) −9.91582e6 −0.508838
\(122\) 6.59472e7 3.28804
\(123\) 8.99651e6 0.435919
\(124\) −4.07482e7 −1.91925
\(125\) 1.03351e7 0.473294
\(126\) 1.89696e7 0.844816
\(127\) −3.33858e7 −1.44627 −0.723134 0.690708i \(-0.757299\pi\)
−0.723134 + 0.690708i \(0.757299\pi\)
\(128\) 7.46368e7 3.14571
\(129\) −1.64422e7 −0.674367
\(130\) 9.67377e6 0.386184
\(131\) 4.37782e7 1.70141 0.850704 0.525645i \(-0.176176\pi\)
0.850704 + 0.525645i \(0.176176\pi\)
\(132\) 2.90560e7 1.09958
\(133\) 3.05228e7 1.12498
\(134\) 9.26292e7 3.32568
\(135\) −8.36404e6 −0.292582
\(136\) −7.97343e7 −2.71806
\(137\) −2.96908e7 −0.986507 −0.493254 0.869886i \(-0.664193\pi\)
−0.493254 + 0.869886i \(0.664193\pi\)
\(138\) 5.81318e7 1.88294
\(139\) −6.45869e6 −0.203983 −0.101991 0.994785i \(-0.532521\pi\)
−0.101991 + 0.994785i \(0.532521\pi\)
\(140\) 1.76323e8 5.43076
\(141\) 2.49446e7 0.749393
\(142\) 1.76674e7 0.517802
\(143\) −3.22868e6 −0.0923314
\(144\) 4.38038e7 1.22248
\(145\) −1.02604e8 −2.79496
\(146\) 2.87643e7 0.764924
\(147\) −1.61847e7 −0.420237
\(148\) 2.55512e7 0.647881
\(149\) 3.19641e6 0.0791608 0.0395804 0.999216i \(-0.487398\pi\)
0.0395804 + 0.999216i \(0.487398\pi\)
\(150\) −6.03384e7 −1.45974
\(151\) −1.51144e7 −0.357250 −0.178625 0.983917i \(-0.557165\pi\)
−0.178625 + 0.983917i \(0.557165\pi\)
\(152\) 1.22708e8 2.83413
\(153\) −1.21206e7 −0.273593
\(154\) −8.05041e7 −1.77621
\(155\) −4.97792e7 −1.07371
\(156\) −9.80138e6 −0.206705
\(157\) 6.70787e7 1.38336 0.691681 0.722203i \(-0.256870\pi\)
0.691681 + 0.722203i \(0.256870\pi\)
\(158\) 1.24335e8 2.50780
\(159\) −1.66822e7 −0.329126
\(160\) 2.96137e8 5.71575
\(161\) −1.17738e8 −2.22344
\(162\) 1.15928e7 0.214232
\(163\) −7.17787e7 −1.29819 −0.649096 0.760707i \(-0.724853\pi\)
−0.649096 + 0.760707i \(0.724853\pi\)
\(164\) −1.15903e8 −2.05183
\(165\) 3.54956e7 0.615150
\(166\) −7.74505e7 −1.31415
\(167\) 6.81851e7 1.13287 0.566437 0.824105i \(-0.308321\pi\)
0.566437 + 0.824105i \(0.308321\pi\)
\(168\) −1.54458e8 −2.51320
\(169\) −6.16594e7 −0.982643
\(170\) −1.54119e8 −2.40593
\(171\) 1.86532e7 0.285276
\(172\) 2.11827e8 3.17418
\(173\) 5.64546e7 0.828968 0.414484 0.910057i \(-0.363962\pi\)
0.414484 + 0.910057i \(0.363962\pi\)
\(174\) 1.42212e8 2.04650
\(175\) 1.22207e8 1.72371
\(176\) −1.85896e8 −2.57026
\(177\) 5.54523e6 0.0751646
\(178\) −2.47593e8 −3.29055
\(179\) 6.74593e7 0.879136 0.439568 0.898209i \(-0.355132\pi\)
0.439568 + 0.898209i \(0.355132\pi\)
\(180\) 1.07755e8 1.37716
\(181\) 2.62245e7 0.328724 0.164362 0.986400i \(-0.447443\pi\)
0.164362 + 0.986400i \(0.447443\pi\)
\(182\) 2.71563e7 0.333903
\(183\) −8.16258e7 −0.984575
\(184\) −4.73331e8 −5.60148
\(185\) 3.12141e7 0.362452
\(186\) 6.89953e7 0.786184
\(187\) 5.14381e7 0.575227
\(188\) −3.21364e8 −3.52732
\(189\) −2.34796e7 −0.252973
\(190\) 2.37182e8 2.50867
\(191\) 1.01202e8 1.05092 0.525461 0.850818i \(-0.323893\pi\)
0.525461 + 0.850818i \(0.323893\pi\)
\(192\) −2.02792e8 −2.06774
\(193\) 9.77806e7 0.979044 0.489522 0.871991i \(-0.337171\pi\)
0.489522 + 0.871991i \(0.337171\pi\)
\(194\) −8.15052e7 −0.801455
\(195\) −1.19737e7 −0.115639
\(196\) 2.08510e8 1.97802
\(197\) −2.53181e7 −0.235939 −0.117969 0.993017i \(-0.537638\pi\)
−0.117969 + 0.993017i \(0.537638\pi\)
\(198\) −4.91979e7 −0.450421
\(199\) 1.47608e8 1.32778 0.663888 0.747832i \(-0.268905\pi\)
0.663888 + 0.747832i \(0.268905\pi\)
\(200\) 4.91298e8 4.34250
\(201\) −1.14651e8 −0.995848
\(202\) 3.95759e7 0.337832
\(203\) −2.88030e8 −2.41658
\(204\) 1.56152e8 1.28778
\(205\) −1.41591e8 −1.14788
\(206\) −4.55571e8 −3.63095
\(207\) −7.19524e7 −0.563831
\(208\) 6.27080e7 0.483171
\(209\) −7.91610e7 −0.599790
\(210\) −2.98552e8 −2.22460
\(211\) −1.08855e7 −0.0797736 −0.0398868 0.999204i \(-0.512700\pi\)
−0.0398868 + 0.999204i \(0.512700\pi\)
\(212\) 2.14918e8 1.54917
\(213\) −2.18677e7 −0.155051
\(214\) 6.62757e7 0.462281
\(215\) 2.58774e8 1.77577
\(216\) −9.43927e7 −0.637310
\(217\) −1.39740e8 −0.928353
\(218\) −1.66962e8 −1.09149
\(219\) −3.56028e7 −0.229050
\(220\) −4.57295e8 −2.89545
\(221\) −1.73515e7 −0.108134
\(222\) −4.32636e7 −0.265392
\(223\) 2.34198e8 1.41421 0.707107 0.707106i \(-0.250000\pi\)
0.707107 + 0.707106i \(0.250000\pi\)
\(224\) 8.31319e8 4.94196
\(225\) 7.46836e7 0.437105
\(226\) 3.21741e7 0.185407
\(227\) 2.70843e8 1.53683 0.768417 0.639949i \(-0.221045\pi\)
0.768417 + 0.639949i \(0.221045\pi\)
\(228\) −2.40311e8 −1.34277
\(229\) 7.25435e7 0.399185 0.199592 0.979879i \(-0.436038\pi\)
0.199592 + 0.979879i \(0.436038\pi\)
\(230\) −9.14903e8 −4.95824
\(231\) 9.96436e7 0.531872
\(232\) −1.15794e9 −6.08805
\(233\) 6.86130e7 0.355354 0.177677 0.984089i \(-0.443142\pi\)
0.177677 + 0.984089i \(0.443142\pi\)
\(234\) 1.65958e7 0.0846727
\(235\) −3.92588e8 −1.97333
\(236\) −7.14399e7 −0.353793
\(237\) −1.53895e8 −0.750941
\(238\) −4.32643e8 −2.08022
\(239\) −1.66795e8 −0.790298 −0.395149 0.918617i \(-0.629307\pi\)
−0.395149 + 0.918617i \(0.629307\pi\)
\(240\) −6.89402e8 −3.21909
\(241\) −1.09566e8 −0.504213 −0.252107 0.967699i \(-0.581123\pi\)
−0.252107 + 0.967699i \(0.581123\pi\)
\(242\) −2.16302e8 −0.981086
\(243\) −1.43489e7 −0.0641500
\(244\) 1.05160e9 4.63430
\(245\) 2.54722e8 1.10659
\(246\) 1.96249e8 0.840492
\(247\) 2.67032e7 0.112752
\(248\) −5.61786e8 −2.33878
\(249\) 9.58640e7 0.393512
\(250\) 2.25449e8 0.912554
\(251\) −2.24011e8 −0.894154 −0.447077 0.894496i \(-0.647535\pi\)
−0.447077 + 0.894496i \(0.647535\pi\)
\(252\) 3.02490e8 1.19072
\(253\) 3.05355e8 1.18545
\(254\) −7.28273e8 −2.78853
\(255\) 1.90760e8 0.720437
\(256\) 6.66733e8 2.48378
\(257\) 4.92524e8 1.80993 0.904964 0.425489i \(-0.139898\pi\)
0.904964 + 0.425489i \(0.139898\pi\)
\(258\) −3.58668e8 −1.30024
\(259\) 8.76245e7 0.313384
\(260\) 1.54258e8 0.544304
\(261\) −1.76022e8 −0.612808
\(262\) 9.54971e8 3.28047
\(263\) −4.27801e8 −1.45010 −0.725049 0.688698i \(-0.758183\pi\)
−0.725049 + 0.688698i \(0.758183\pi\)
\(264\) 4.00588e8 1.33994
\(265\) 2.62551e8 0.866669
\(266\) 6.65819e8 2.16905
\(267\) 3.06457e8 0.985327
\(268\) 1.47707e9 4.68736
\(269\) 1.72598e8 0.540633 0.270316 0.962772i \(-0.412872\pi\)
0.270316 + 0.962772i \(0.412872\pi\)
\(270\) −1.82452e8 −0.564125
\(271\) 2.54060e8 0.775431 0.387716 0.921779i \(-0.373264\pi\)
0.387716 + 0.921779i \(0.373264\pi\)
\(272\) −9.99038e8 −3.01017
\(273\) −3.36126e7 −0.0999844
\(274\) −6.47671e8 −1.90208
\(275\) −3.16945e8 −0.919009
\(276\) 9.26971e8 2.65390
\(277\) −3.42400e8 −0.967954 −0.483977 0.875081i \(-0.660808\pi\)
−0.483977 + 0.875081i \(0.660808\pi\)
\(278\) −1.40889e8 −0.393297
\(279\) −8.53986e7 −0.235416
\(280\) 2.43092e9 6.61786
\(281\) −6.48885e8 −1.74460 −0.872300 0.488971i \(-0.837372\pi\)
−0.872300 + 0.488971i \(0.837372\pi\)
\(282\) 5.44138e8 1.44490
\(283\) −1.70974e8 −0.448413 −0.224206 0.974542i \(-0.571979\pi\)
−0.224206 + 0.974542i \(0.571979\pi\)
\(284\) 2.81725e8 0.729812
\(285\) −2.93571e8 −0.751201
\(286\) −7.04300e7 −0.178023
\(287\) −3.97475e8 −0.992482
\(288\) 5.08038e8 1.25321
\(289\) −1.33902e8 −0.326320
\(290\) −2.23818e9 −5.38894
\(291\) 1.00883e8 0.239989
\(292\) 4.58676e8 1.07812
\(293\) −7.65359e8 −1.77758 −0.888789 0.458316i \(-0.848453\pi\)
−0.888789 + 0.458316i \(0.848453\pi\)
\(294\) −3.53051e8 −0.810255
\(295\) −8.72732e7 −0.197926
\(296\) 3.52268e8 0.789501
\(297\) 6.08945e7 0.134875
\(298\) 6.97260e7 0.152629
\(299\) −1.03005e8 −0.222847
\(300\) −9.62157e8 −2.05741
\(301\) 7.26433e8 1.53537
\(302\) −3.29704e8 −0.688811
\(303\) −4.89849e7 −0.101161
\(304\) 1.53748e9 3.13871
\(305\) 1.28466e9 2.59262
\(306\) −2.64398e8 −0.527513
\(307\) 1.89113e8 0.373024 0.186512 0.982453i \(-0.440282\pi\)
0.186512 + 0.982453i \(0.440282\pi\)
\(308\) −1.28372e9 −2.50347
\(309\) 5.63880e8 1.08726
\(310\) −1.08588e9 −2.07021
\(311\) 2.41634e8 0.455508 0.227754 0.973719i \(-0.426862\pi\)
0.227754 + 0.973719i \(0.426862\pi\)
\(312\) −1.35129e8 −0.251889
\(313\) −4.22149e8 −0.778145 −0.389073 0.921207i \(-0.627204\pi\)
−0.389073 + 0.921207i \(0.627204\pi\)
\(314\) 1.46325e9 2.66725
\(315\) 3.69531e8 0.666138
\(316\) 1.98265e9 3.53461
\(317\) 1.02142e9 1.80093 0.900467 0.434925i \(-0.143225\pi\)
0.900467 + 0.434925i \(0.143225\pi\)
\(318\) −3.63903e8 −0.634586
\(319\) 7.47008e8 1.28842
\(320\) 3.19162e9 5.44486
\(321\) −8.20324e7 −0.138426
\(322\) −2.56832e9 −4.28700
\(323\) −4.25425e8 −0.702448
\(324\) 1.84859e8 0.301948
\(325\) 1.06914e8 0.172760
\(326\) −1.56577e9 −2.50303
\(327\) 2.06656e8 0.326837
\(328\) −1.59793e9 −2.50034
\(329\) −1.10208e9 −1.70618
\(330\) 7.74297e8 1.18607
\(331\) 8.62048e8 1.30657 0.653286 0.757111i \(-0.273390\pi\)
0.653286 + 0.757111i \(0.273390\pi\)
\(332\) −1.23503e9 −1.85222
\(333\) 5.35493e7 0.0794693
\(334\) 1.48738e9 2.18428
\(335\) 1.80443e9 2.62231
\(336\) −1.93529e9 −2.78330
\(337\) −9.09270e8 −1.29416 −0.647080 0.762422i \(-0.724010\pi\)
−0.647080 + 0.762422i \(0.724010\pi\)
\(338\) −1.34503e9 −1.89462
\(339\) −3.98233e7 −0.0555186
\(340\) −2.45758e9 −3.39103
\(341\) 3.62418e8 0.494959
\(342\) 4.06897e8 0.550039
\(343\) −2.67336e8 −0.357708
\(344\) 2.92041e9 3.86803
\(345\) 1.13242e9 1.48470
\(346\) 1.23149e9 1.59833
\(347\) 4.58759e8 0.589428 0.294714 0.955585i \(-0.404776\pi\)
0.294714 + 0.955585i \(0.404776\pi\)
\(348\) 2.26771e9 2.88443
\(349\) −9.57438e8 −1.20565 −0.602826 0.797873i \(-0.705959\pi\)
−0.602826 + 0.797873i \(0.705959\pi\)
\(350\) 2.66581e9 3.32346
\(351\) −2.05414e7 −0.0253545
\(352\) −2.15603e9 −2.63485
\(353\) 3.43926e8 0.416154 0.208077 0.978112i \(-0.433280\pi\)
0.208077 + 0.978112i \(0.433280\pi\)
\(354\) 1.20963e8 0.144924
\(355\) 3.44164e8 0.408287
\(356\) −3.94812e9 −4.63784
\(357\) 5.35501e8 0.622905
\(358\) 1.47155e9 1.69505
\(359\) −9.03951e8 −1.03113 −0.515566 0.856850i \(-0.672418\pi\)
−0.515566 + 0.856850i \(0.672418\pi\)
\(360\) 1.48559e9 1.67819
\(361\) −2.39161e8 −0.267556
\(362\) 5.72057e8 0.633810
\(363\) 2.67727e8 0.293778
\(364\) 4.33034e8 0.470617
\(365\) 5.60332e8 0.603144
\(366\) −1.78057e9 −1.89835
\(367\) 1.55115e9 1.63804 0.819019 0.573767i \(-0.194518\pi\)
0.819019 + 0.573767i \(0.194518\pi\)
\(368\) −5.93065e9 −6.20347
\(369\) −2.42906e8 −0.251678
\(370\) 6.80900e8 0.698840
\(371\) 7.37035e8 0.749341
\(372\) 1.10020e9 1.10808
\(373\) −1.60756e9 −1.60393 −0.801967 0.597368i \(-0.796213\pi\)
−0.801967 + 0.597368i \(0.796213\pi\)
\(374\) 1.12206e9 1.10909
\(375\) −2.79049e8 −0.273257
\(376\) −4.43057e9 −4.29836
\(377\) −2.51987e8 −0.242205
\(378\) −5.12180e8 −0.487755
\(379\) 5.29474e8 0.499583 0.249791 0.968300i \(-0.419638\pi\)
0.249791 + 0.968300i \(0.419638\pi\)
\(380\) 3.78211e9 3.53583
\(381\) 9.01416e8 0.835003
\(382\) 2.20760e9 2.02627
\(383\) −1.13814e9 −1.03515 −0.517573 0.855639i \(-0.673164\pi\)
−0.517573 + 0.855639i \(0.673164\pi\)
\(384\) −2.01519e9 −1.81617
\(385\) −1.56823e9 −1.40055
\(386\) 2.13297e9 1.88769
\(387\) 4.43940e8 0.389346
\(388\) −1.29968e9 −1.12961
\(389\) −8.63651e8 −0.743900 −0.371950 0.928253i \(-0.621311\pi\)
−0.371950 + 0.928253i \(0.621311\pi\)
\(390\) −2.61192e8 −0.222963
\(391\) 1.64103e9 1.38834
\(392\) 2.87467e9 2.41039
\(393\) −1.18201e9 −0.982308
\(394\) −5.52285e8 −0.454911
\(395\) 2.42206e9 1.97741
\(396\) −7.84511e8 −0.634842
\(397\) 3.54481e7 0.0284332 0.0142166 0.999899i \(-0.495475\pi\)
0.0142166 + 0.999899i \(0.495475\pi\)
\(398\) 3.21991e9 2.56007
\(399\) −8.24114e8 −0.649505
\(400\) 6.15576e9 4.80919
\(401\) 2.17381e8 0.168351 0.0841756 0.996451i \(-0.473174\pi\)
0.0841756 + 0.996451i \(0.473174\pi\)
\(402\) −2.50099e9 −1.92008
\(403\) −1.22254e8 −0.0930453
\(404\) 6.31078e8 0.476155
\(405\) 2.25829e8 0.168922
\(406\) −6.28305e9 −4.65939
\(407\) −2.27255e8 −0.167083
\(408\) 2.15283e9 1.56927
\(409\) 2.43499e9 1.75981 0.879904 0.475151i \(-0.157607\pi\)
0.879904 + 0.475151i \(0.157607\pi\)
\(410\) −3.08864e9 −2.21322
\(411\) 8.01652e8 0.569560
\(412\) −7.26454e9 −5.11762
\(413\) −2.44994e8 −0.171131
\(414\) −1.56956e9 −1.08712
\(415\) −1.50875e9 −1.03621
\(416\) 7.27290e8 0.495314
\(417\) 1.74385e8 0.117769
\(418\) −1.72681e9 −1.15645
\(419\) −4.11950e8 −0.273587 −0.136794 0.990600i \(-0.543680\pi\)
−0.136794 + 0.990600i \(0.543680\pi\)
\(420\) −4.76071e9 −3.13545
\(421\) −1.54001e9 −1.00585 −0.502927 0.864329i \(-0.667744\pi\)
−0.502927 + 0.864329i \(0.667744\pi\)
\(422\) −2.37454e8 −0.153811
\(423\) −6.73504e8 −0.432662
\(424\) 2.96303e9 1.88780
\(425\) −1.70332e9 −1.07630
\(426\) −4.77020e8 −0.298953
\(427\) 3.60631e9 2.24164
\(428\) 1.05683e9 0.651559
\(429\) 8.71744e7 0.0533075
\(430\) 5.64487e9 3.42385
\(431\) 2.03052e9 1.22162 0.610811 0.791776i \(-0.290843\pi\)
0.610811 + 0.791776i \(0.290843\pi\)
\(432\) −1.18270e9 −0.705801
\(433\) 6.40897e8 0.379385 0.189693 0.981844i \(-0.439251\pi\)
0.189693 + 0.981844i \(0.439251\pi\)
\(434\) −3.04828e9 −1.78995
\(435\) 2.77030e9 1.61367
\(436\) −2.66238e9 −1.53839
\(437\) −2.52547e9 −1.44763
\(438\) −7.76635e8 −0.441629
\(439\) −4.52255e8 −0.255128 −0.127564 0.991830i \(-0.540716\pi\)
−0.127564 + 0.991830i \(0.540716\pi\)
\(440\) −6.30462e9 −3.52837
\(441\) 4.36987e8 0.242624
\(442\) −3.78503e8 −0.208493
\(443\) 1.95166e9 1.06657 0.533287 0.845935i \(-0.320957\pi\)
0.533287 + 0.845935i \(0.320957\pi\)
\(444\) −6.89882e8 −0.374054
\(445\) −4.82315e9 −2.59460
\(446\) 5.10875e9 2.72673
\(447\) −8.63030e7 −0.0457035
\(448\) 8.95954e9 4.70774
\(449\) −7.36502e8 −0.383983 −0.191991 0.981397i \(-0.561495\pi\)
−0.191991 + 0.981397i \(0.561495\pi\)
\(450\) 1.62914e9 0.842779
\(451\) 1.03085e9 0.529150
\(452\) 5.13049e8 0.261321
\(453\) 4.08090e8 0.206259
\(454\) 5.90813e9 2.96316
\(455\) 5.29008e8 0.263283
\(456\) −3.31311e9 −1.63628
\(457\) 8.44943e8 0.414115 0.207057 0.978329i \(-0.433611\pi\)
0.207057 + 0.978329i \(0.433611\pi\)
\(458\) 1.58245e9 0.769664
\(459\) 3.27257e8 0.157959
\(460\) −1.45891e10 −6.98835
\(461\) 2.59377e8 0.123305 0.0616523 0.998098i \(-0.480363\pi\)
0.0616523 + 0.998098i \(0.480363\pi\)
\(462\) 2.17361e9 1.02550
\(463\) 4.01914e8 0.188191 0.0940957 0.995563i \(-0.470004\pi\)
0.0940957 + 0.995563i \(0.470004\pi\)
\(464\) −1.45085e10 −6.74233
\(465\) 1.34404e9 0.619907
\(466\) 1.49671e9 0.685154
\(467\) −3.10097e9 −1.40893 −0.704463 0.709740i \(-0.748812\pi\)
−0.704463 + 0.709740i \(0.748812\pi\)
\(468\) 2.64637e8 0.119341
\(469\) 5.06541e9 2.26730
\(470\) −8.56386e9 −3.80476
\(471\) −1.81113e9 −0.798685
\(472\) −9.84925e8 −0.431128
\(473\) −1.88401e9 −0.818596
\(474\) −3.35705e9 −1.44788
\(475\) 2.62133e9 1.12226
\(476\) −6.89893e9 −2.93195
\(477\) 4.50419e8 0.190021
\(478\) −3.63844e9 −1.52377
\(479\) 3.79632e9 1.57829 0.789147 0.614204i \(-0.210523\pi\)
0.789147 + 0.614204i \(0.210523\pi\)
\(480\) −7.99571e9 −3.29999
\(481\) 7.66594e7 0.0314092
\(482\) −2.39005e9 −0.972169
\(483\) 3.17893e9 1.28371
\(484\) −3.44916e9 −1.38278
\(485\) −1.58773e9 −0.631948
\(486\) −3.13005e8 −0.123687
\(487\) 7.82358e6 0.00306940 0.00153470 0.999999i \(-0.499511\pi\)
0.00153470 + 0.999999i \(0.499511\pi\)
\(488\) 1.44981e10 5.64731
\(489\) 1.93802e9 0.749511
\(490\) 5.55646e9 2.13360
\(491\) −2.48479e9 −0.947336 −0.473668 0.880704i \(-0.657070\pi\)
−0.473668 + 0.880704i \(0.657070\pi\)
\(492\) 3.12938e9 1.18463
\(493\) 4.01455e9 1.50894
\(494\) 5.82500e8 0.217396
\(495\) −9.58382e8 −0.355157
\(496\) −7.03894e9 −2.59013
\(497\) 9.66138e8 0.353014
\(498\) 2.09116e9 0.758727
\(499\) 3.32153e9 1.19670 0.598352 0.801234i \(-0.295823\pi\)
0.598352 + 0.801234i \(0.295823\pi\)
\(500\) 3.59502e9 1.28619
\(501\) −1.84100e9 −0.654065
\(502\) −4.88655e9 −1.72401
\(503\) 2.45772e9 0.861082 0.430541 0.902571i \(-0.358323\pi\)
0.430541 + 0.902571i \(0.358323\pi\)
\(504\) 4.17036e9 1.45100
\(505\) 7.70945e8 0.266381
\(506\) 6.66096e9 2.28565
\(507\) 1.66480e9 0.567329
\(508\) −1.16131e10 −3.93028
\(509\) −4.91205e8 −0.165101 −0.0825507 0.996587i \(-0.526307\pi\)
−0.0825507 + 0.996587i \(0.526307\pi\)
\(510\) 4.16120e9 1.38907
\(511\) 1.57297e9 0.521491
\(512\) 4.99051e9 1.64324
\(513\) −5.03635e8 −0.164704
\(514\) 1.07438e10 3.48970
\(515\) −8.87458e9 −2.86301
\(516\) −5.71933e9 −1.83262
\(517\) 2.85824e9 0.909667
\(518\) 1.91143e9 0.604232
\(519\) −1.52427e9 −0.478605
\(520\) 2.12672e9 0.663283
\(521\) 3.12235e9 0.967275 0.483638 0.875268i \(-0.339315\pi\)
0.483638 + 0.875268i \(0.339315\pi\)
\(522\) −3.83971e9 −1.18155
\(523\) 1.24953e9 0.381938 0.190969 0.981596i \(-0.438837\pi\)
0.190969 + 0.981596i \(0.438837\pi\)
\(524\) 1.52280e10 4.62363
\(525\) −3.29959e9 −0.995182
\(526\) −9.33200e9 −2.79592
\(527\) 1.94770e9 0.579674
\(528\) 5.01920e9 1.48394
\(529\) 6.33690e9 1.86115
\(530\) 5.72725e9 1.67102
\(531\) −1.49721e8 −0.0433963
\(532\) 1.06172e10 3.05716
\(533\) −3.47736e8 −0.0994727
\(534\) 6.68501e9 1.89980
\(535\) 1.29106e9 0.364509
\(536\) 2.03640e10 5.71197
\(537\) −1.82140e9 −0.507569
\(538\) 3.76502e9 1.04239
\(539\) −1.85451e9 −0.510114
\(540\) −2.90938e9 −0.795101
\(541\) 8.58513e8 0.233108 0.116554 0.993184i \(-0.462815\pi\)
0.116554 + 0.993184i \(0.462815\pi\)
\(542\) 5.54202e9 1.49510
\(543\) −7.08061e8 −0.189789
\(544\) −1.15869e10 −3.08582
\(545\) −3.25244e9 −0.860641
\(546\) −7.33219e8 −0.192779
\(547\) 6.72132e9 1.75590 0.877948 0.478755i \(-0.158912\pi\)
0.877948 + 0.478755i \(0.158912\pi\)
\(548\) −1.03278e10 −2.68087
\(549\) 2.20390e9 0.568445
\(550\) −6.91379e9 −1.77193
\(551\) −6.17823e9 −1.57338
\(552\) 1.27799e10 3.23402
\(553\) 6.79923e9 1.70971
\(554\) −7.46907e9 −1.86630
\(555\) −8.42781e8 −0.209262
\(556\) −2.24662e9 −0.554329
\(557\) −2.12425e9 −0.520850 −0.260425 0.965494i \(-0.583863\pi\)
−0.260425 + 0.965494i \(0.583863\pi\)
\(558\) −1.86287e9 −0.453903
\(559\) 6.35529e8 0.153884
\(560\) 3.04584e10 7.32908
\(561\) −1.38883e9 −0.332107
\(562\) −1.41547e10 −3.36375
\(563\) 1.69196e9 0.399587 0.199793 0.979838i \(-0.435973\pi\)
0.199793 + 0.979838i \(0.435973\pi\)
\(564\) 8.67683e9 2.03650
\(565\) 6.26756e8 0.146194
\(566\) −3.72961e9 −0.864580
\(567\) 6.33949e8 0.146054
\(568\) 3.88407e9 0.889341
\(569\) −7.54396e8 −0.171675 −0.0858374 0.996309i \(-0.527357\pi\)
−0.0858374 + 0.996309i \(0.527357\pi\)
\(570\) −6.40392e9 −1.44838
\(571\) −3.56448e9 −0.801254 −0.400627 0.916241i \(-0.631208\pi\)
−0.400627 + 0.916241i \(0.631208\pi\)
\(572\) −1.12308e9 −0.250913
\(573\) −2.73244e9 −0.606750
\(574\) −8.67045e9 −1.91360
\(575\) −1.01115e10 −2.21808
\(576\) 5.47538e9 1.19381
\(577\) −6.20845e9 −1.34545 −0.672725 0.739892i \(-0.734876\pi\)
−0.672725 + 0.739892i \(0.734876\pi\)
\(578\) −2.92091e9 −0.629175
\(579\) −2.64008e9 −0.565251
\(580\) −3.56901e10 −7.59539
\(581\) −4.23537e9 −0.895931
\(582\) 2.20064e9 0.462720
\(583\) −1.91151e9 −0.399517
\(584\) 6.32366e9 1.31378
\(585\) 3.23289e8 0.0667645
\(586\) −1.66954e10 −3.42733
\(587\) −1.55164e9 −0.316635 −0.158317 0.987388i \(-0.550607\pi\)
−0.158317 + 0.987388i \(0.550607\pi\)
\(588\) −5.62976e9 −1.14201
\(589\) −2.99742e9 −0.604428
\(590\) −1.90376e9 −0.381620
\(591\) 6.83589e8 0.136219
\(592\) 4.41378e9 0.874349
\(593\) 6.42351e9 1.26497 0.632486 0.774572i \(-0.282035\pi\)
0.632486 + 0.774572i \(0.282035\pi\)
\(594\) 1.32834e9 0.260051
\(595\) −8.42794e9 −1.64026
\(596\) 1.11185e9 0.215122
\(597\) −3.98543e9 −0.766592
\(598\) −2.24693e9 −0.429670
\(599\) −3.92988e9 −0.747111 −0.373556 0.927608i \(-0.621861\pi\)
−0.373556 + 0.927608i \(0.621861\pi\)
\(600\) −1.32650e10 −2.50714
\(601\) −5.05280e9 −0.949448 −0.474724 0.880135i \(-0.657452\pi\)
−0.474724 + 0.880135i \(0.657452\pi\)
\(602\) 1.58463e10 2.96033
\(603\) 3.09559e9 0.574953
\(604\) −5.25747e9 −0.970839
\(605\) −4.21360e9 −0.773588
\(606\) −1.06855e9 −0.195048
\(607\) −7.19598e9 −1.30596 −0.652979 0.757376i \(-0.726481\pi\)
−0.652979 + 0.757376i \(0.726481\pi\)
\(608\) 1.78317e10 3.21759
\(609\) 7.77681e9 1.39521
\(610\) 2.80234e10 4.99881
\(611\) −9.64165e8 −0.171004
\(612\) −4.21609e9 −0.743499
\(613\) −4.01935e9 −0.704765 −0.352383 0.935856i \(-0.614628\pi\)
−0.352383 + 0.935856i \(0.614628\pi\)
\(614\) 4.12528e9 0.719223
\(615\) 3.82295e9 0.662729
\(616\) −1.76984e10 −3.05071
\(617\) 2.76235e9 0.473458 0.236729 0.971576i \(-0.423925\pi\)
0.236729 + 0.971576i \(0.423925\pi\)
\(618\) 1.23004e10 2.09633
\(619\) 6.94825e9 1.17749 0.588746 0.808318i \(-0.299622\pi\)
0.588746 + 0.808318i \(0.299622\pi\)
\(620\) −1.73154e10 −2.91784
\(621\) 1.94272e9 0.325528
\(622\) 5.27096e9 0.878261
\(623\) −1.35396e10 −2.24335
\(624\) −1.69312e9 −0.278959
\(625\) −3.61185e9 −0.591766
\(626\) −9.20870e9 −1.50033
\(627\) 2.13735e9 0.346289
\(628\) 2.33329e10 3.75933
\(629\) −1.22131e9 −0.195680
\(630\) 8.06090e9 1.28437
\(631\) 4.84723e9 0.768052 0.384026 0.923322i \(-0.374537\pi\)
0.384026 + 0.923322i \(0.374537\pi\)
\(632\) 2.73343e10 4.30723
\(633\) 2.93908e8 0.0460573
\(634\) 2.22811e10 3.47236
\(635\) −1.41869e10 −2.19876
\(636\) −5.80280e9 −0.894412
\(637\) 6.25576e8 0.0958942
\(638\) 1.62951e10 2.48419
\(639\) 5.90429e8 0.0895189
\(640\) 3.17160e10 4.78242
\(641\) −1.40292e9 −0.210393 −0.105196 0.994451i \(-0.533547\pi\)
−0.105196 + 0.994451i \(0.533547\pi\)
\(642\) −1.78944e9 −0.266898
\(643\) −8.62484e9 −1.27942 −0.639709 0.768617i \(-0.720945\pi\)
−0.639709 + 0.768617i \(0.720945\pi\)
\(644\) −4.09545e10 −6.04228
\(645\) −6.98691e9 −1.02524
\(646\) −9.28015e9 −1.35438
\(647\) −2.56483e9 −0.372300 −0.186150 0.982521i \(-0.559601\pi\)
−0.186150 + 0.982521i \(0.559601\pi\)
\(648\) 2.54860e9 0.367951
\(649\) 6.35393e8 0.0912402
\(650\) 2.33221e9 0.333098
\(651\) 3.77299e9 0.535985
\(652\) −2.49678e10 −3.52788
\(653\) −2.05512e9 −0.288829 −0.144414 0.989517i \(-0.546130\pi\)
−0.144414 + 0.989517i \(0.546130\pi\)
\(654\) 4.50797e9 0.630172
\(655\) 1.86030e10 2.58665
\(656\) −2.00214e10 −2.76905
\(657\) 9.61277e8 0.132242
\(658\) −2.40405e10 −3.28968
\(659\) −7.82229e9 −1.06472 −0.532360 0.846518i \(-0.678695\pi\)
−0.532360 + 0.846518i \(0.678695\pi\)
\(660\) 1.23470e10 1.67169
\(661\) −4.59162e8 −0.0618388 −0.0309194 0.999522i \(-0.509844\pi\)
−0.0309194 + 0.999522i \(0.509844\pi\)
\(662\) 1.88046e10 2.51919
\(663\) 4.68490e8 0.0624314
\(664\) −1.70270e10 −2.25710
\(665\) 1.29703e10 1.71030
\(666\) 1.16812e9 0.153224
\(667\) 2.38318e10 3.10968
\(668\) 2.37178e10 3.07862
\(669\) −6.32333e9 −0.816497
\(670\) 3.93616e10 5.05604
\(671\) −9.35299e9 −1.19515
\(672\) −2.24456e10 −2.85324
\(673\) −6.96032e9 −0.880190 −0.440095 0.897951i \(-0.645055\pi\)
−0.440095 + 0.897951i \(0.645055\pi\)
\(674\) −1.98347e10 −2.49526
\(675\) −2.01646e9 −0.252363
\(676\) −2.14479e10 −2.67036
\(677\) 1.09451e10 1.35568 0.677842 0.735208i \(-0.262915\pi\)
0.677842 + 0.735208i \(0.262915\pi\)
\(678\) −8.68700e8 −0.107045
\(679\) −4.45710e9 −0.546396
\(680\) −3.38821e10 −4.13227
\(681\) −7.31276e9 −0.887292
\(682\) 7.90574e9 0.954326
\(683\) 2.98731e9 0.358763 0.179382 0.983780i \(-0.442590\pi\)
0.179382 + 0.983780i \(0.442590\pi\)
\(684\) 6.48839e9 0.775248
\(685\) −1.26167e10 −1.49979
\(686\) −5.83164e9 −0.689693
\(687\) −1.95867e9 −0.230469
\(688\) 3.65916e10 4.28372
\(689\) 6.44804e8 0.0751036
\(690\) 2.47024e10 2.86264
\(691\) 5.37053e9 0.619219 0.309609 0.950864i \(-0.399802\pi\)
0.309609 + 0.950864i \(0.399802\pi\)
\(692\) 1.96374e10 2.25275
\(693\) −2.69038e9 −0.307077
\(694\) 1.00073e10 1.13647
\(695\) −2.74454e9 −0.310115
\(696\) 3.12644e10 3.51494
\(697\) 5.53998e9 0.619717
\(698\) −2.08854e10 −2.32461
\(699\) −1.85255e9 −0.205163
\(700\) 4.25090e10 4.68423
\(701\) 6.64856e9 0.728979 0.364489 0.931208i \(-0.381244\pi\)
0.364489 + 0.931208i \(0.381244\pi\)
\(702\) −4.48087e8 −0.0488858
\(703\) 1.87954e9 0.204036
\(704\) −2.32366e10 −2.50997
\(705\) 1.05999e10 1.13930
\(706\) 7.50236e9 0.802382
\(707\) 2.16420e9 0.230319
\(708\) 1.92888e9 0.204262
\(709\) 5.41047e8 0.0570129 0.0285064 0.999594i \(-0.490925\pi\)
0.0285064 + 0.999594i \(0.490925\pi\)
\(710\) 7.50753e9 0.787215
\(711\) 4.15517e9 0.433556
\(712\) −5.44319e10 −5.65163
\(713\) 1.15622e10 1.19461
\(714\) 1.16814e10 1.20102
\(715\) −1.37199e9 −0.140372
\(716\) 2.34653e10 2.38908
\(717\) 4.50347e9 0.456278
\(718\) −1.97186e10 −1.98812
\(719\) 9.95307e9 0.998633 0.499316 0.866420i \(-0.333585\pi\)
0.499316 + 0.866420i \(0.333585\pi\)
\(720\) 1.86139e10 1.85854
\(721\) −2.49128e10 −2.47542
\(722\) −5.21702e9 −0.515872
\(723\) 2.95827e9 0.291108
\(724\) 9.12203e9 0.893319
\(725\) −2.47364e10 −2.41076
\(726\) 5.84016e9 0.566430
\(727\) −1.32542e10 −1.27933 −0.639665 0.768654i \(-0.720927\pi\)
−0.639665 + 0.768654i \(0.720927\pi\)
\(728\) 5.97015e9 0.573489
\(729\) 3.87420e8 0.0370370
\(730\) 1.22230e10 1.16292
\(731\) −1.01250e10 −0.958703
\(732\) −2.83931e10 −2.67561
\(733\) 6.98110e8 0.0654727 0.0327363 0.999464i \(-0.489578\pi\)
0.0327363 + 0.999464i \(0.489578\pi\)
\(734\) 3.38367e10 3.15828
\(735\) −6.87749e9 −0.638887
\(736\) −6.87838e10 −6.35937
\(737\) −1.31372e10 −1.20883
\(738\) −5.29871e9 −0.485258
\(739\) 5.01583e9 0.457180 0.228590 0.973523i \(-0.426588\pi\)
0.228590 + 0.973523i \(0.426588\pi\)
\(740\) 1.08576e10 0.984974
\(741\) −7.20987e8 −0.0650974
\(742\) 1.60776e10 1.44480
\(743\) 1.65206e10 1.47763 0.738814 0.673910i \(-0.235386\pi\)
0.738814 + 0.673910i \(0.235386\pi\)
\(744\) 1.51682e10 1.35030
\(745\) 1.35827e9 0.120348
\(746\) −3.50671e10 −3.09253
\(747\) −2.58833e9 −0.227194
\(748\) 1.78924e10 1.56320
\(749\) 3.62427e9 0.315162
\(750\) −6.08713e9 −0.526863
\(751\) 1.65574e10 1.42643 0.713216 0.700944i \(-0.247238\pi\)
0.713216 + 0.700944i \(0.247238\pi\)
\(752\) −5.55133e10 −4.76030
\(753\) 6.04831e9 0.516240
\(754\) −5.49680e9 −0.466993
\(755\) −6.42269e9 −0.543128
\(756\) −8.16723e9 −0.687462
\(757\) −8.37645e9 −0.701818 −0.350909 0.936410i \(-0.614127\pi\)
−0.350909 + 0.936410i \(0.614127\pi\)
\(758\) 1.15499e10 0.963240
\(759\) −8.24458e9 −0.684419
\(760\) 5.21431e10 4.30873
\(761\) 2.25820e9 0.185745 0.0928725 0.995678i \(-0.470395\pi\)
0.0928725 + 0.995678i \(0.470395\pi\)
\(762\) 1.96634e10 1.60996
\(763\) −9.13028e9 −0.744129
\(764\) 3.52024e10 2.85592
\(765\) −5.15051e9 −0.415944
\(766\) −2.48273e10 −1.99585
\(767\) −2.14336e8 −0.0171519
\(768\) −1.80018e10 −1.43401
\(769\) 5.84510e9 0.463500 0.231750 0.972775i \(-0.425555\pi\)
0.231750 + 0.972775i \(0.425555\pi\)
\(770\) −3.42092e10 −2.70038
\(771\) −1.32981e10 −1.04496
\(772\) 3.40124e10 2.66058
\(773\) −4.05810e9 −0.316005 −0.158003 0.987439i \(-0.550505\pi\)
−0.158003 + 0.987439i \(0.550505\pi\)
\(774\) 9.68404e9 0.750695
\(775\) −1.20011e10 −0.926115
\(776\) −1.79184e10 −1.37653
\(777\) −2.36586e9 −0.180932
\(778\) −1.88396e10 −1.43431
\(779\) −8.52580e9 −0.646181
\(780\) −4.16497e9 −0.314254
\(781\) −2.50569e9 −0.188212
\(782\) 3.57971e10 2.67685
\(783\) 4.75259e9 0.353805
\(784\) 3.60185e10 2.66943
\(785\) 2.85042e10 2.10313
\(786\) −2.57842e10 −1.89398
\(787\) −1.54824e10 −1.13221 −0.566106 0.824333i \(-0.691551\pi\)
−0.566106 + 0.824333i \(0.691551\pi\)
\(788\) −8.80675e9 −0.641171
\(789\) 1.15506e10 0.837214
\(790\) 5.28346e10 3.81262
\(791\) 1.75943e9 0.126402
\(792\) −1.08159e10 −0.773612
\(793\) 3.15502e9 0.224671
\(794\) 7.73260e8 0.0548218
\(795\) −7.08888e9 −0.500371
\(796\) 5.13447e10 3.60828
\(797\) 2.07763e10 1.45366 0.726831 0.686816i \(-0.240992\pi\)
0.726831 + 0.686816i \(0.240992\pi\)
\(798\) −1.79771e10 −1.25230
\(799\) 1.53607e10 1.06536
\(800\) 7.13947e10 4.93005
\(801\) −8.27434e9 −0.568879
\(802\) 4.74192e9 0.324596
\(803\) −4.07951e9 −0.278037
\(804\) −3.98808e10 −2.70625
\(805\) −5.00313e10 −3.38031
\(806\) −2.66682e9 −0.179400
\(807\) −4.66014e9 −0.312134
\(808\) 8.70053e9 0.580238
\(809\) 1.24188e10 0.824628 0.412314 0.911042i \(-0.364721\pi\)
0.412314 + 0.911042i \(0.364721\pi\)
\(810\) 4.92620e9 0.325698
\(811\) −2.86382e10 −1.88526 −0.942632 0.333835i \(-0.891657\pi\)
−0.942632 + 0.333835i \(0.891657\pi\)
\(812\) −1.00190e11 −6.56714
\(813\) −6.85961e9 −0.447695
\(814\) −4.95730e9 −0.322151
\(815\) −3.05014e10 −1.97364
\(816\) 2.69740e10 1.73792
\(817\) 1.55819e10 0.999642
\(818\) 5.31165e10 3.39307
\(819\) 9.07539e8 0.0577260
\(820\) −4.92515e10 −3.11940
\(821\) 6.53708e9 0.412271 0.206135 0.978523i \(-0.433911\pi\)
0.206135 + 0.978523i \(0.433911\pi\)
\(822\) 1.74871e10 1.09816
\(823\) −6.03479e8 −0.0377366 −0.0188683 0.999822i \(-0.506006\pi\)
−0.0188683 + 0.999822i \(0.506006\pi\)
\(824\) −1.00155e11 −6.23628
\(825\) 8.55752e9 0.530590
\(826\) −5.34426e9 −0.329957
\(827\) −1.88036e10 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(828\) −2.50282e10 −1.53223
\(829\) 1.67240e10 1.01953 0.509763 0.860315i \(-0.329733\pi\)
0.509763 + 0.860315i \(0.329733\pi\)
\(830\) −3.29116e10 −1.99791
\(831\) 9.24480e9 0.558849
\(832\) 7.83836e9 0.471839
\(833\) −9.96642e9 −0.597423
\(834\) 3.80400e9 0.227070
\(835\) 2.89744e10 1.72231
\(836\) −2.75357e10 −1.62995
\(837\) 2.30576e9 0.135918
\(838\) −8.98622e9 −0.527501
\(839\) 2.23616e10 1.30718 0.653592 0.756847i \(-0.273261\pi\)
0.653592 + 0.756847i \(0.273261\pi\)
\(840\) −6.56349e10 −3.82083
\(841\) 4.10514e10 2.37981
\(842\) −3.35935e10 −1.93938
\(843\) 1.75199e10 1.00725
\(844\) −3.78645e9 −0.216787
\(845\) −2.62014e10 −1.49391
\(846\) −1.46917e10 −0.834212
\(847\) −1.18284e10 −0.668861
\(848\) 3.71256e10 2.09068
\(849\) 4.61630e9 0.258891
\(850\) −3.71559e10 −2.07521
\(851\) −7.25011e9 −0.403265
\(852\) −7.60657e9 −0.421357
\(853\) −1.44921e9 −0.0799483 −0.0399742 0.999201i \(-0.512728\pi\)
−0.0399742 + 0.999201i \(0.512728\pi\)
\(854\) 7.86675e10 4.32208
\(855\) 7.92642e9 0.433706
\(856\) 1.45703e10 0.793983
\(857\) −8.02175e9 −0.435347 −0.217674 0.976022i \(-0.569847\pi\)
−0.217674 + 0.976022i \(0.569847\pi\)
\(858\) 1.90161e9 0.102782
\(859\) 1.19930e10 0.645584 0.322792 0.946470i \(-0.395379\pi\)
0.322792 + 0.946470i \(0.395379\pi\)
\(860\) 9.00132e10 4.82571
\(861\) 1.07318e10 0.573010
\(862\) 4.42935e10 2.35540
\(863\) −2.67440e10 −1.41641 −0.708205 0.706007i \(-0.750495\pi\)
−0.708205 + 0.706007i \(0.750495\pi\)
\(864\) −1.37170e10 −0.723539
\(865\) 2.39896e10 1.26028
\(866\) 1.39804e10 0.731489
\(867\) 3.61535e9 0.188401
\(868\) −4.86079e10 −2.52283
\(869\) −1.76339e10 −0.911546
\(870\) 6.04310e10 3.11130
\(871\) 4.43153e9 0.227243
\(872\) −3.67056e10 −1.87467
\(873\) −2.72383e9 −0.138558
\(874\) −5.50903e10 −2.79116
\(875\) 1.23286e10 0.622139
\(876\) −1.23842e10 −0.622451
\(877\) −6.68932e8 −0.0334875 −0.0167438 0.999860i \(-0.505330\pi\)
−0.0167438 + 0.999860i \(0.505330\pi\)
\(878\) −9.86542e9 −0.491909
\(879\) 2.06647e10 1.02629
\(880\) −7.89942e10 −3.90756
\(881\) 2.73982e10 1.34992 0.674958 0.737856i \(-0.264162\pi\)
0.674958 + 0.737856i \(0.264162\pi\)
\(882\) 9.53238e9 0.467801
\(883\) 1.56749e9 0.0766198 0.0383099 0.999266i \(-0.487803\pi\)
0.0383099 + 0.999266i \(0.487803\pi\)
\(884\) −6.03561e9 −0.293859
\(885\) 2.35638e9 0.114273
\(886\) 4.25732e10 2.05645
\(887\) 6.07648e9 0.292361 0.146181 0.989258i \(-0.453302\pi\)
0.146181 + 0.989258i \(0.453302\pi\)
\(888\) −9.51125e9 −0.455819
\(889\) −3.98254e10 −1.90110
\(890\) −1.05211e11 −5.00263
\(891\) −1.64415e9 −0.0778699
\(892\) 8.14643e10 3.84317
\(893\) −2.36395e10 −1.11086
\(894\) −1.88260e9 −0.0881205
\(895\) 2.86659e10 1.33655
\(896\) 8.90332e10 4.13499
\(897\) 2.78112e9 0.128661
\(898\) −1.60659e10 −0.740354
\(899\) 2.82854e10 1.29838
\(900\) 2.59782e10 1.18785
\(901\) −1.02728e10 −0.467897
\(902\) 2.24869e10 1.02025
\(903\) −1.96137e10 −0.886446
\(904\) 7.07329e9 0.318443
\(905\) 1.11438e10 0.499760
\(906\) 8.90201e9 0.397685
\(907\) −3.19626e10 −1.42238 −0.711191 0.702999i \(-0.751844\pi\)
−0.711191 + 0.702999i \(0.751844\pi\)
\(908\) 9.42112e10 4.17640
\(909\) 1.32259e9 0.0584053
\(910\) 1.15397e10 0.507633
\(911\) 3.33824e10 1.46286 0.731430 0.681916i \(-0.238853\pi\)
0.731430 + 0.681916i \(0.238853\pi\)
\(912\) −4.15119e10 −1.81214
\(913\) 1.09845e10 0.477673
\(914\) 1.84315e10 0.798451
\(915\) −3.46859e10 −1.49685
\(916\) 2.52338e10 1.08480
\(917\) 5.22224e10 2.23648
\(918\) 7.13874e9 0.304560
\(919\) −8.98427e9 −0.381837 −0.190919 0.981606i \(-0.561147\pi\)
−0.190919 + 0.981606i \(0.561147\pi\)
\(920\) −2.01136e11 −8.51594
\(921\) −5.10604e9 −0.215365
\(922\) 5.65802e9 0.237742
\(923\) 8.45238e8 0.0353812
\(924\) 3.46604e10 1.44538
\(925\) 7.52530e9 0.312628
\(926\) 8.76730e9 0.362850
\(927\) −1.52248e10 −0.627728
\(928\) −1.68270e11 −6.91177
\(929\) −4.29286e10 −1.75668 −0.878339 0.478039i \(-0.841348\pi\)
−0.878339 + 0.478039i \(0.841348\pi\)
\(930\) 2.93187e10 1.19524
\(931\) 1.53379e10 0.622934
\(932\) 2.38666e10 0.965685
\(933\) −6.52411e9 −0.262988
\(934\) −6.76441e10 −2.71654
\(935\) 2.18579e10 0.874518
\(936\) 3.64849e9 0.145428
\(937\) −2.92971e9 −0.116342 −0.0581710 0.998307i \(-0.518527\pi\)
−0.0581710 + 0.998307i \(0.518527\pi\)
\(938\) 1.10496e11 4.37156
\(939\) 1.13980e10 0.449262
\(940\) −1.36560e11 −5.36259
\(941\) 1.23877e10 0.484650 0.242325 0.970195i \(-0.422090\pi\)
0.242325 + 0.970195i \(0.422090\pi\)
\(942\) −3.95076e10 −1.53994
\(943\) 3.28873e10 1.27714
\(944\) −1.23407e10 −0.477461
\(945\) −9.97734e9 −0.384595
\(946\) −4.10975e10 −1.57833
\(947\) 5.27697e9 0.201911 0.100955 0.994891i \(-0.467810\pi\)
0.100955 + 0.994891i \(0.467810\pi\)
\(948\) −5.35315e10 −2.04071
\(949\) 1.37613e9 0.0522671
\(950\) 5.71814e10 2.16382
\(951\) −2.75784e10 −1.03977
\(952\) −9.51139e10 −3.57285
\(953\) −1.49617e10 −0.559959 −0.279980 0.960006i \(-0.590328\pi\)
−0.279980 + 0.960006i \(0.590328\pi\)
\(954\) 9.82537e9 0.366378
\(955\) 4.30043e10 1.59772
\(956\) −5.80187e10 −2.14766
\(957\) −2.01692e10 −0.743871
\(958\) 8.28123e10 3.04309
\(959\) −3.54178e10 −1.29675
\(960\) −8.61738e10 −3.14359
\(961\) −1.37897e10 −0.501213
\(962\) 1.67224e9 0.0605599
\(963\) 2.21488e9 0.0799204
\(964\) −3.81117e10 −1.37022
\(965\) 4.15506e10 1.48844
\(966\) 6.93446e10 2.47510
\(967\) 3.49771e10 1.24392 0.621958 0.783051i \(-0.286337\pi\)
0.621958 + 0.783051i \(0.286337\pi\)
\(968\) −4.75528e10 −1.68505
\(969\) 1.14865e10 0.405558
\(970\) −3.46346e10 −1.21845
\(971\) 4.12202e10 1.44492 0.722459 0.691414i \(-0.243012\pi\)
0.722459 + 0.691414i \(0.243012\pi\)
\(972\) −4.99118e9 −0.174330
\(973\) −7.70449e9 −0.268132
\(974\) 1.70662e8 0.00591809
\(975\) −2.88669e9 −0.0997433
\(976\) 1.81655e11 6.25423
\(977\) −3.74910e10 −1.28616 −0.643081 0.765798i \(-0.722344\pi\)
−0.643081 + 0.765798i \(0.722344\pi\)
\(978\) 4.22758e10 1.44513
\(979\) 3.51150e10 1.19606
\(980\) 8.86035e10 3.00718
\(981\) −5.57972e9 −0.188700
\(982\) −5.42028e10 −1.82655
\(983\) 5.07263e9 0.170332 0.0851660 0.996367i \(-0.472858\pi\)
0.0851660 + 0.996367i \(0.472858\pi\)
\(984\) 4.31441e10 1.44357
\(985\) −1.07586e10 −0.358698
\(986\) 8.75728e10 2.90938
\(987\) 2.97561e10 0.985066
\(988\) 9.28856e9 0.306407
\(989\) −6.01056e10 −1.97573
\(990\) −2.09060e10 −0.684775
\(991\) 1.45130e10 0.473694 0.236847 0.971547i \(-0.423886\pi\)
0.236847 + 0.971547i \(0.423886\pi\)
\(992\) −8.16379e10 −2.65522
\(993\) −2.32753e10 −0.754350
\(994\) 2.10752e10 0.680643
\(995\) 6.27243e10 2.01862
\(996\) 3.33457e10 1.06938
\(997\) 1.97848e10 0.632265 0.316132 0.948715i \(-0.397616\pi\)
0.316132 + 0.948715i \(0.397616\pi\)
\(998\) 7.24554e10 2.30735
\(999\) −1.44583e9 −0.0458816
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.c.1.17 17
3.2 odd 2 531.8.a.c.1.1 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.17 17 1.1 even 1 trivial
531.8.a.c.1.1 17 3.2 odd 2