Properties

Label 177.8.a.d.1.9
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 1.9
Root \(2.80127\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.80127 q^{2} +27.0000 q^{3} -113.550 q^{4} -87.8737 q^{5} +102.634 q^{6} +1306.06 q^{7} -918.197 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+3.80127 q^{2} +27.0000 q^{3} -113.550 q^{4} -87.8737 q^{5} +102.634 q^{6} +1306.06 q^{7} -918.197 q^{8} +729.000 q^{9} -334.031 q^{10} -3799.92 q^{11} -3065.86 q^{12} +8976.87 q^{13} +4964.66 q^{14} -2372.59 q^{15} +11044.1 q^{16} -4357.97 q^{17} +2771.12 q^{18} -57085.5 q^{19} +9978.10 q^{20} +35263.5 q^{21} -14444.5 q^{22} +96669.6 q^{23} -24791.3 q^{24} -70403.2 q^{25} +34123.5 q^{26} +19683.0 q^{27} -148303. q^{28} +22753.4 q^{29} -9018.85 q^{30} -53358.6 q^{31} +159511. q^{32} -102598. q^{33} -16565.8 q^{34} -114768. q^{35} -82778.2 q^{36} +319784. q^{37} -216997. q^{38} +242376. q^{39} +80685.4 q^{40} +476338. q^{41} +134046. q^{42} +707558. q^{43} +431483. q^{44} -64060.0 q^{45} +367467. q^{46} +946330. q^{47} +298192. q^{48} +882238. q^{49} -267621. q^{50} -117665. q^{51} -1.01933e6 q^{52} +799926. q^{53} +74820.3 q^{54} +333913. q^{55} -1.19922e6 q^{56} -1.54131e6 q^{57} +86491.9 q^{58} +205379. q^{59} +269409. q^{60} +539847. q^{61} -202830. q^{62} +952114. q^{63} -807306. q^{64} -788832. q^{65} -390002. q^{66} +2.90460e6 q^{67} +494849. q^{68} +2.61008e6 q^{69} -436264. q^{70} -523802. q^{71} -669366. q^{72} -2.11143e6 q^{73} +1.21558e6 q^{74} -1.90089e6 q^{75} +6.48208e6 q^{76} -4.96291e6 q^{77} +921334. q^{78} -3.80223e6 q^{79} -970490. q^{80} +531441. q^{81} +1.81069e6 q^{82} +1.88881e6 q^{83} -4.00418e6 q^{84} +382951. q^{85} +2.68962e6 q^{86} +614343. q^{87} +3.48908e6 q^{88} +6.48074e6 q^{89} -243509. q^{90} +1.17243e7 q^{91} -1.09769e7 q^{92} -1.44068e6 q^{93} +3.59725e6 q^{94} +5.01631e6 q^{95} +4.30680e6 q^{96} +2.84275e6 q^{97} +3.35362e6 q^{98} -2.77014e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.80127 0.335988 0.167994 0.985788i \(-0.446271\pi\)
0.167994 + 0.985788i \(0.446271\pi\)
\(3\) 27.0000 0.577350
\(4\) −113.550 −0.887112
\(5\) −87.8737 −0.314387 −0.157193 0.987568i \(-0.550245\pi\)
−0.157193 + 0.987568i \(0.550245\pi\)
\(6\) 102.634 0.193983
\(7\) 1306.06 1.43919 0.719596 0.694393i \(-0.244327\pi\)
0.719596 + 0.694393i \(0.244327\pi\)
\(8\) −918.197 −0.634046
\(9\) 729.000 0.333333
\(10\) −334.031 −0.105630
\(11\) −3799.92 −0.860796 −0.430398 0.902639i \(-0.641627\pi\)
−0.430398 + 0.902639i \(0.641627\pi\)
\(12\) −3065.86 −0.512175
\(13\) 8976.87 1.13324 0.566622 0.823978i \(-0.308250\pi\)
0.566622 + 0.823978i \(0.308250\pi\)
\(14\) 4964.66 0.483550
\(15\) −2372.59 −0.181511
\(16\) 11044.1 0.674081
\(17\) −4357.97 −0.215136 −0.107568 0.994198i \(-0.534306\pi\)
−0.107568 + 0.994198i \(0.534306\pi\)
\(18\) 2771.12 0.111996
\(19\) −57085.5 −1.90936 −0.954680 0.297633i \(-0.903803\pi\)
−0.954680 + 0.297633i \(0.903803\pi\)
\(20\) 9978.10 0.278896
\(21\) 35263.5 0.830917
\(22\) −14444.5 −0.289217
\(23\) 96669.6 1.65670 0.828348 0.560214i \(-0.189281\pi\)
0.828348 + 0.560214i \(0.189281\pi\)
\(24\) −24791.3 −0.366067
\(25\) −70403.2 −0.901161
\(26\) 34123.5 0.380756
\(27\) 19683.0 0.192450
\(28\) −148303. −1.27672
\(29\) 22753.4 0.173242 0.0866212 0.996241i \(-0.472393\pi\)
0.0866212 + 0.996241i \(0.472393\pi\)
\(30\) −9018.85 −0.0609855
\(31\) −53358.6 −0.321691 −0.160845 0.986980i \(-0.551422\pi\)
−0.160845 + 0.986980i \(0.551422\pi\)
\(32\) 159511. 0.860529
\(33\) −102598. −0.496981
\(34\) −16565.8 −0.0722830
\(35\) −114768. −0.452462
\(36\) −82778.2 −0.295704
\(37\) 319784. 1.03789 0.518944 0.854808i \(-0.326325\pi\)
0.518944 + 0.854808i \(0.326325\pi\)
\(38\) −216997. −0.641522
\(39\) 242376. 0.654279
\(40\) 80685.4 0.199336
\(41\) 476338. 1.07937 0.539687 0.841866i \(-0.318543\pi\)
0.539687 + 0.841866i \(0.318543\pi\)
\(42\) 134046. 0.279178
\(43\) 707558. 1.35713 0.678567 0.734539i \(-0.262601\pi\)
0.678567 + 0.734539i \(0.262601\pi\)
\(44\) 431483. 0.763623
\(45\) −64060.0 −0.104796
\(46\) 367467. 0.556629
\(47\) 946330. 1.32954 0.664768 0.747050i \(-0.268530\pi\)
0.664768 + 0.747050i \(0.268530\pi\)
\(48\) 298192. 0.389181
\(49\) 882238. 1.07127
\(50\) −267621. −0.302779
\(51\) −117665. −0.124209
\(52\) −1.01933e6 −1.00531
\(53\) 799926. 0.738047 0.369024 0.929420i \(-0.379692\pi\)
0.369024 + 0.929420i \(0.379692\pi\)
\(54\) 74820.3 0.0646608
\(55\) 333913. 0.270623
\(56\) −1.19922e6 −0.912514
\(57\) −1.54131e6 −1.10237
\(58\) 86491.9 0.0582073
\(59\) 205379. 0.130189
\(60\) 269409. 0.161021
\(61\) 539847. 0.304520 0.152260 0.988340i \(-0.451345\pi\)
0.152260 + 0.988340i \(0.451345\pi\)
\(62\) −202830. −0.108084
\(63\) 952114. 0.479730
\(64\) −807306. −0.384954
\(65\) −788832. −0.356277
\(66\) −390002. −0.166979
\(67\) 2.90460e6 1.17985 0.589923 0.807460i \(-0.299158\pi\)
0.589923 + 0.807460i \(0.299158\pi\)
\(68\) 494849. 0.190850
\(69\) 2.61008e6 0.956493
\(70\) −436264. −0.152022
\(71\) −523802. −0.173685 −0.0868426 0.996222i \(-0.527678\pi\)
−0.0868426 + 0.996222i \(0.527678\pi\)
\(72\) −669366. −0.211349
\(73\) −2.11143e6 −0.635252 −0.317626 0.948216i \(-0.602886\pi\)
−0.317626 + 0.948216i \(0.602886\pi\)
\(74\) 1.21558e6 0.348718
\(75\) −1.90089e6 −0.520286
\(76\) 6.48208e6 1.69382
\(77\) −4.96291e6 −1.23885
\(78\) 921334. 0.219830
\(79\) −3.80223e6 −0.867648 −0.433824 0.900998i \(-0.642836\pi\)
−0.433824 + 0.900998i \(0.642836\pi\)
\(80\) −970490. −0.211922
\(81\) 531441. 0.111111
\(82\) 1.81069e6 0.362656
\(83\) 1.88881e6 0.362589 0.181294 0.983429i \(-0.441971\pi\)
0.181294 + 0.983429i \(0.441971\pi\)
\(84\) −4.00418e6 −0.737117
\(85\) 382951. 0.0676359
\(86\) 2.68962e6 0.455980
\(87\) 614343. 0.100022
\(88\) 3.48908e6 0.545785
\(89\) 6.48074e6 0.974450 0.487225 0.873277i \(-0.338009\pi\)
0.487225 + 0.873277i \(0.338009\pi\)
\(90\) −243509. −0.0352100
\(91\) 1.17243e7 1.63095
\(92\) −1.09769e7 −1.46967
\(93\) −1.44068e6 −0.185728
\(94\) 3.59725e6 0.446708
\(95\) 5.01631e6 0.600278
\(96\) 4.30680e6 0.496827
\(97\) 2.84275e6 0.316256 0.158128 0.987419i \(-0.449454\pi\)
0.158128 + 0.987419i \(0.449454\pi\)
\(98\) 3.35362e6 0.359934
\(99\) −2.77014e6 −0.286932
\(100\) 7.99431e6 0.799431
\(101\) 1.41130e6 0.136299 0.0681497 0.997675i \(-0.478290\pi\)
0.0681497 + 0.997675i \(0.478290\pi\)
\(102\) −447277. −0.0417326
\(103\) 1.05972e7 0.955564 0.477782 0.878478i \(-0.341441\pi\)
0.477782 + 0.878478i \(0.341441\pi\)
\(104\) −8.24254e6 −0.718529
\(105\) −3.09874e6 −0.261229
\(106\) 3.04073e6 0.247975
\(107\) 1.84524e7 1.45616 0.728082 0.685490i \(-0.240412\pi\)
0.728082 + 0.685490i \(0.240412\pi\)
\(108\) −2.23501e6 −0.170725
\(109\) −1.18614e7 −0.877289 −0.438645 0.898661i \(-0.644541\pi\)
−0.438645 + 0.898661i \(0.644541\pi\)
\(110\) 1.26929e6 0.0909259
\(111\) 8.63417e6 0.599225
\(112\) 1.44243e7 0.970131
\(113\) 1.57496e7 1.02682 0.513412 0.858142i \(-0.328381\pi\)
0.513412 + 0.858142i \(0.328381\pi\)
\(114\) −5.85892e6 −0.370383
\(115\) −8.49472e6 −0.520843
\(116\) −2.58366e6 −0.153685
\(117\) 6.54414e6 0.377748
\(118\) 780700. 0.0437419
\(119\) −5.69175e6 −0.309622
\(120\) 2.17851e6 0.115087
\(121\) −5.04777e6 −0.259030
\(122\) 2.05210e6 0.102315
\(123\) 1.28611e7 0.623177
\(124\) 6.05889e6 0.285376
\(125\) 1.30517e7 0.597700
\(126\) 3.61924e6 0.161183
\(127\) −1.74843e7 −0.757417 −0.378708 0.925516i \(-0.623632\pi\)
−0.378708 + 0.925516i \(0.623632\pi\)
\(128\) −2.34862e7 −0.989869
\(129\) 1.91041e7 0.783541
\(130\) −2.99856e6 −0.119705
\(131\) 3.83994e7 1.49236 0.746182 0.665742i \(-0.231885\pi\)
0.746182 + 0.665742i \(0.231885\pi\)
\(132\) 1.16500e7 0.440878
\(133\) −7.45568e7 −2.74794
\(134\) 1.10412e7 0.396413
\(135\) −1.72962e6 −0.0605037
\(136\) 4.00148e6 0.136406
\(137\) −4.77564e7 −1.58675 −0.793376 0.608731i \(-0.791679\pi\)
−0.793376 + 0.608731i \(0.791679\pi\)
\(138\) 9.92161e6 0.321370
\(139\) −9.01121e6 −0.284598 −0.142299 0.989824i \(-0.545449\pi\)
−0.142299 + 0.989824i \(0.545449\pi\)
\(140\) 1.30319e7 0.401385
\(141\) 2.55509e7 0.767608
\(142\) −1.99111e6 −0.0583561
\(143\) −3.41114e7 −0.975492
\(144\) 8.05118e6 0.224694
\(145\) −1.99943e6 −0.0544651
\(146\) −8.02609e6 −0.213437
\(147\) 2.38204e7 0.618499
\(148\) −3.63116e7 −0.920724
\(149\) −1.73883e7 −0.430631 −0.215315 0.976545i \(-0.569078\pi\)
−0.215315 + 0.976545i \(0.569078\pi\)
\(150\) −7.22577e6 −0.174809
\(151\) 1.46672e7 0.346680 0.173340 0.984862i \(-0.444544\pi\)
0.173340 + 0.984862i \(0.444544\pi\)
\(152\) 5.24157e7 1.21062
\(153\) −3.17696e6 −0.0717120
\(154\) −1.88653e7 −0.416238
\(155\) 4.68882e6 0.101135
\(156\) −2.75218e7 −0.580419
\(157\) 4.11300e7 0.848222 0.424111 0.905610i \(-0.360587\pi\)
0.424111 + 0.905610i \(0.360587\pi\)
\(158\) −1.44533e7 −0.291519
\(159\) 2.15980e7 0.426112
\(160\) −1.40168e7 −0.270539
\(161\) 1.26256e8 2.38430
\(162\) 2.02015e6 0.0373320
\(163\) −1.02265e6 −0.0184957 −0.00924783 0.999957i \(-0.502944\pi\)
−0.00924783 + 0.999957i \(0.502944\pi\)
\(164\) −5.40884e7 −0.957526
\(165\) 9.01566e6 0.156244
\(166\) 7.17986e6 0.121825
\(167\) −4.54084e6 −0.0754447 −0.0377224 0.999288i \(-0.512010\pi\)
−0.0377224 + 0.999288i \(0.512010\pi\)
\(168\) −3.23788e7 −0.526840
\(169\) 1.78358e7 0.284242
\(170\) 1.45570e6 0.0227248
\(171\) −4.16153e7 −0.636454
\(172\) −8.03435e7 −1.20393
\(173\) 1.45643e7 0.213860 0.106930 0.994267i \(-0.465898\pi\)
0.106930 + 0.994267i \(0.465898\pi\)
\(174\) 2.33528e6 0.0336060
\(175\) −9.19505e7 −1.29694
\(176\) −4.19669e7 −0.580246
\(177\) 5.54523e6 0.0751646
\(178\) 2.46350e7 0.327403
\(179\) −7.59717e7 −0.990071 −0.495036 0.868873i \(-0.664845\pi\)
−0.495036 + 0.868873i \(0.664845\pi\)
\(180\) 7.27403e6 0.0929654
\(181\) −1.08062e8 −1.35456 −0.677281 0.735724i \(-0.736842\pi\)
−0.677281 + 0.735724i \(0.736842\pi\)
\(182\) 4.45672e7 0.547981
\(183\) 1.45759e7 0.175815
\(184\) −8.87618e7 −1.05042
\(185\) −2.81006e7 −0.326298
\(186\) −5.47642e6 −0.0624024
\(187\) 1.65599e7 0.185188
\(188\) −1.07456e8 −1.17945
\(189\) 2.57071e7 0.276972
\(190\) 1.90683e7 0.201686
\(191\) −1.32489e8 −1.37583 −0.687914 0.725792i \(-0.741473\pi\)
−0.687914 + 0.725792i \(0.741473\pi\)
\(192\) −2.17973e7 −0.222253
\(193\) 1.01108e8 1.01236 0.506178 0.862429i \(-0.331058\pi\)
0.506178 + 0.862429i \(0.331058\pi\)
\(194\) 1.08061e7 0.106258
\(195\) −2.12985e7 −0.205696
\(196\) −1.00178e8 −0.950338
\(197\) 1.00275e7 0.0934464 0.0467232 0.998908i \(-0.485122\pi\)
0.0467232 + 0.998908i \(0.485122\pi\)
\(198\) −1.05300e7 −0.0964056
\(199\) 6.35124e7 0.571312 0.285656 0.958332i \(-0.407789\pi\)
0.285656 + 0.958332i \(0.407789\pi\)
\(200\) 6.46440e7 0.571378
\(201\) 7.84243e7 0.681184
\(202\) 5.36472e6 0.0457949
\(203\) 2.97173e7 0.249329
\(204\) 1.33609e7 0.110187
\(205\) −4.18576e7 −0.339341
\(206\) 4.02827e7 0.321058
\(207\) 7.04722e7 0.552232
\(208\) 9.91418e7 0.763898
\(209\) 2.16920e8 1.64357
\(210\) −1.17791e7 −0.0877698
\(211\) 2.91443e7 0.213582 0.106791 0.994281i \(-0.465942\pi\)
0.106791 + 0.994281i \(0.465942\pi\)
\(212\) −9.08319e7 −0.654731
\(213\) −1.41427e7 −0.100277
\(214\) 7.01426e7 0.489253
\(215\) −6.21758e7 −0.426665
\(216\) −1.80729e7 −0.122022
\(217\) −6.96893e7 −0.462975
\(218\) −4.50883e7 −0.294758
\(219\) −5.70085e7 −0.366763
\(220\) −3.79160e7 −0.240073
\(221\) −3.91210e7 −0.243802
\(222\) 3.28208e7 0.201332
\(223\) −2.58995e8 −1.56396 −0.781979 0.623305i \(-0.785789\pi\)
−0.781979 + 0.623305i \(0.785789\pi\)
\(224\) 2.08330e8 1.23847
\(225\) −5.13239e7 −0.300387
\(226\) 5.98686e7 0.345000
\(227\) −2.65429e8 −1.50612 −0.753058 0.657955i \(-0.771422\pi\)
−0.753058 + 0.657955i \(0.771422\pi\)
\(228\) 1.75016e8 0.977926
\(229\) 1.83433e8 1.00938 0.504688 0.863302i \(-0.331607\pi\)
0.504688 + 0.863302i \(0.331607\pi\)
\(230\) −3.22907e7 −0.174997
\(231\) −1.33999e8 −0.715250
\(232\) −2.08921e7 −0.109844
\(233\) 3.14430e8 1.62846 0.814232 0.580540i \(-0.197159\pi\)
0.814232 + 0.580540i \(0.197159\pi\)
\(234\) 2.48760e7 0.126919
\(235\) −8.31576e7 −0.417989
\(236\) −2.33209e7 −0.115492
\(237\) −1.02660e8 −0.500937
\(238\) −2.16359e7 −0.104029
\(239\) −3.42402e7 −0.162235 −0.0811173 0.996705i \(-0.525849\pi\)
−0.0811173 + 0.996705i \(0.525849\pi\)
\(240\) −2.62032e7 −0.122353
\(241\) −1.98928e8 −0.915452 −0.457726 0.889093i \(-0.651336\pi\)
−0.457726 + 0.889093i \(0.651336\pi\)
\(242\) −1.91879e7 −0.0870309
\(243\) 1.43489e7 0.0641500
\(244\) −6.12999e7 −0.270144
\(245\) −7.75255e7 −0.336793
\(246\) 4.88886e7 0.209380
\(247\) −5.12449e8 −2.16377
\(248\) 4.89938e7 0.203967
\(249\) 5.09978e7 0.209341
\(250\) 4.96131e7 0.200820
\(251\) −4.06923e8 −1.62425 −0.812127 0.583481i \(-0.801690\pi\)
−0.812127 + 0.583481i \(0.801690\pi\)
\(252\) −1.08113e8 −0.425575
\(253\) −3.67337e8 −1.42608
\(254\) −6.64624e7 −0.254483
\(255\) 1.03397e7 0.0390496
\(256\) 1.40580e7 0.0523700
\(257\) 1.04442e8 0.383804 0.191902 0.981414i \(-0.438534\pi\)
0.191902 + 0.981414i \(0.438534\pi\)
\(258\) 7.26196e7 0.263260
\(259\) 4.17656e8 1.49372
\(260\) 8.95721e7 0.316058
\(261\) 1.65873e7 0.0577475
\(262\) 1.45966e8 0.501415
\(263\) 2.20493e8 0.747393 0.373696 0.927551i \(-0.378090\pi\)
0.373696 + 0.927551i \(0.378090\pi\)
\(264\) 9.42051e7 0.315109
\(265\) −7.02925e7 −0.232032
\(266\) −2.83410e8 −0.923272
\(267\) 1.74980e8 0.562599
\(268\) −3.29819e8 −1.04666
\(269\) 3.53895e8 1.10852 0.554258 0.832345i \(-0.313002\pi\)
0.554258 + 0.832345i \(0.313002\pi\)
\(270\) −6.57474e6 −0.0203285
\(271\) −4.86543e8 −1.48501 −0.742504 0.669842i \(-0.766362\pi\)
−0.742504 + 0.669842i \(0.766362\pi\)
\(272\) −4.81300e7 −0.145019
\(273\) 3.16556e8 0.941632
\(274\) −1.81535e8 −0.533129
\(275\) 2.67527e8 0.775716
\(276\) −2.96376e8 −0.848517
\(277\) −4.24477e8 −1.19998 −0.599992 0.800006i \(-0.704829\pi\)
−0.599992 + 0.800006i \(0.704829\pi\)
\(278\) −3.42540e7 −0.0956213
\(279\) −3.88984e7 −0.107230
\(280\) 1.05380e8 0.286882
\(281\) 5.85274e8 1.57357 0.786787 0.617225i \(-0.211743\pi\)
0.786787 + 0.617225i \(0.211743\pi\)
\(282\) 9.71258e7 0.257907
\(283\) 1.23529e8 0.323979 0.161989 0.986793i \(-0.448209\pi\)
0.161989 + 0.986793i \(0.448209\pi\)
\(284\) 5.94779e7 0.154078
\(285\) 1.35440e8 0.346570
\(286\) −1.29667e8 −0.327753
\(287\) 6.22124e8 1.55343
\(288\) 1.16283e8 0.286843
\(289\) −3.91347e8 −0.953716
\(290\) −7.60036e6 −0.0182996
\(291\) 7.67544e7 0.182590
\(292\) 2.39753e8 0.563540
\(293\) −3.34796e8 −0.777577 −0.388788 0.921327i \(-0.627106\pi\)
−0.388788 + 0.921327i \(0.627106\pi\)
\(294\) 9.05477e7 0.207808
\(295\) −1.80474e7 −0.0409297
\(296\) −2.93625e8 −0.658069
\(297\) −7.47939e7 −0.165660
\(298\) −6.60975e7 −0.144687
\(299\) 8.67791e8 1.87744
\(300\) 2.15846e8 0.461552
\(301\) 9.24110e8 1.95317
\(302\) 5.57541e7 0.116480
\(303\) 3.81051e7 0.0786925
\(304\) −6.30460e8 −1.28706
\(305\) −4.74384e7 −0.0957372
\(306\) −1.20765e7 −0.0240943
\(307\) 5.54710e8 1.09416 0.547081 0.837080i \(-0.315739\pi\)
0.547081 + 0.837080i \(0.315739\pi\)
\(308\) 5.63540e8 1.09900
\(309\) 2.86124e8 0.551695
\(310\) 1.78235e7 0.0339802
\(311\) −4.01552e8 −0.756973 −0.378486 0.925607i \(-0.623555\pi\)
−0.378486 + 0.925607i \(0.623555\pi\)
\(312\) −2.22549e8 −0.414843
\(313\) 7.36811e8 1.35816 0.679080 0.734064i \(-0.262379\pi\)
0.679080 + 0.734064i \(0.262379\pi\)
\(314\) 1.56346e8 0.284992
\(315\) −8.36659e7 −0.150821
\(316\) 4.31745e8 0.769701
\(317\) −2.52797e7 −0.0445722 −0.0222861 0.999752i \(-0.507094\pi\)
−0.0222861 + 0.999752i \(0.507094\pi\)
\(318\) 8.20997e7 0.143168
\(319\) −8.64613e7 −0.149126
\(320\) 7.09410e7 0.121024
\(321\) 4.98216e8 0.840717
\(322\) 4.79932e8 0.801096
\(323\) 2.48777e8 0.410772
\(324\) −6.03453e7 −0.0985680
\(325\) −6.32001e8 −1.02124
\(326\) −3.88736e6 −0.00621431
\(327\) −3.20258e8 −0.506503
\(328\) −4.37372e8 −0.684373
\(329\) 1.23596e9 1.91346
\(330\) 3.42709e7 0.0524961
\(331\) 6.96332e8 1.05540 0.527702 0.849430i \(-0.323054\pi\)
0.527702 + 0.849430i \(0.323054\pi\)
\(332\) −2.14475e8 −0.321657
\(333\) 2.33123e8 0.345963
\(334\) −1.72609e7 −0.0253485
\(335\) −2.55238e8 −0.370928
\(336\) 3.89455e8 0.560105
\(337\) −7.32502e8 −1.04257 −0.521284 0.853384i \(-0.674547\pi\)
−0.521284 + 0.853384i \(0.674547\pi\)
\(338\) 6.77984e7 0.0955017
\(339\) 4.25241e8 0.592838
\(340\) −4.34843e7 −0.0600006
\(341\) 2.02759e8 0.276910
\(342\) −1.58191e8 −0.213841
\(343\) 7.66587e7 0.102573
\(344\) −6.49678e8 −0.860485
\(345\) −2.29357e8 −0.300709
\(346\) 5.53629e7 0.0718543
\(347\) −6.58184e8 −0.845657 −0.422828 0.906210i \(-0.638963\pi\)
−0.422828 + 0.906210i \(0.638963\pi\)
\(348\) −6.97589e7 −0.0887303
\(349\) −3.82277e7 −0.0481381 −0.0240691 0.999710i \(-0.507662\pi\)
−0.0240691 + 0.999710i \(0.507662\pi\)
\(350\) −3.49528e8 −0.435757
\(351\) 1.76692e8 0.218093
\(352\) −6.06129e8 −0.740740
\(353\) 8.56374e8 1.03622 0.518110 0.855314i \(-0.326636\pi\)
0.518110 + 0.855314i \(0.326636\pi\)
\(354\) 2.10789e7 0.0252544
\(355\) 4.60284e7 0.0546043
\(356\) −7.35890e8 −0.864446
\(357\) −1.53677e8 −0.178760
\(358\) −2.88789e8 −0.332652
\(359\) 1.50693e9 1.71895 0.859475 0.511177i \(-0.170790\pi\)
0.859475 + 0.511177i \(0.170790\pi\)
\(360\) 5.88197e7 0.0664452
\(361\) 2.36488e9 2.64566
\(362\) −4.10773e8 −0.455116
\(363\) −1.36290e8 −0.149551
\(364\) −1.33130e9 −1.44684
\(365\) 1.85539e8 0.199715
\(366\) 5.54068e7 0.0590716
\(367\) −7.69693e8 −0.812805 −0.406402 0.913694i \(-0.633217\pi\)
−0.406402 + 0.913694i \(0.633217\pi\)
\(368\) 1.06763e9 1.11675
\(369\) 3.47251e8 0.359791
\(370\) −1.06818e8 −0.109632
\(371\) 1.04475e9 1.06219
\(372\) 1.63590e8 0.164762
\(373\) −8.69970e8 −0.868008 −0.434004 0.900911i \(-0.642900\pi\)
−0.434004 + 0.900911i \(0.642900\pi\)
\(374\) 6.29488e7 0.0622209
\(375\) 3.52397e8 0.345082
\(376\) −8.68918e8 −0.842988
\(377\) 2.04255e8 0.196326
\(378\) 9.77195e7 0.0930593
\(379\) −6.86916e8 −0.648136 −0.324068 0.946034i \(-0.605051\pi\)
−0.324068 + 0.946034i \(0.605051\pi\)
\(380\) −5.69604e8 −0.532514
\(381\) −4.72076e8 −0.437295
\(382\) −5.03627e8 −0.462261
\(383\) 6.92766e8 0.630073 0.315037 0.949080i \(-0.397983\pi\)
0.315037 + 0.949080i \(0.397983\pi\)
\(384\) −6.34127e8 −0.571501
\(385\) 4.36109e8 0.389478
\(386\) 3.84337e8 0.340139
\(387\) 5.15810e8 0.452378
\(388\) −3.22796e8 −0.280554
\(389\) −1.80412e7 −0.0155397 −0.00776985 0.999970i \(-0.502473\pi\)
−0.00776985 + 0.999970i \(0.502473\pi\)
\(390\) −8.09611e7 −0.0691115
\(391\) −4.21283e8 −0.356415
\(392\) −8.10068e8 −0.679236
\(393\) 1.03678e9 0.861616
\(394\) 3.81174e7 0.0313968
\(395\) 3.34116e8 0.272777
\(396\) 3.14551e8 0.254541
\(397\) 6.47110e8 0.519053 0.259526 0.965736i \(-0.416434\pi\)
0.259526 + 0.965736i \(0.416434\pi\)
\(398\) 2.41428e8 0.191954
\(399\) −2.01303e9 −1.58652
\(400\) −7.77543e8 −0.607455
\(401\) −2.37807e8 −0.184171 −0.0920853 0.995751i \(-0.529353\pi\)
−0.0920853 + 0.995751i \(0.529353\pi\)
\(402\) 2.98112e8 0.228869
\(403\) −4.78994e8 −0.364554
\(404\) −1.60253e8 −0.120913
\(405\) −4.66997e7 −0.0349319
\(406\) 1.12963e8 0.0837714
\(407\) −1.21515e9 −0.893410
\(408\) 1.08040e8 0.0787542
\(409\) 1.17282e9 0.847618 0.423809 0.905752i \(-0.360693\pi\)
0.423809 + 0.905752i \(0.360693\pi\)
\(410\) −1.59112e8 −0.114014
\(411\) −1.28942e9 −0.916112
\(412\) −1.20331e9 −0.847693
\(413\) 2.68236e8 0.187367
\(414\) 2.67883e8 0.185543
\(415\) −1.65977e8 −0.113993
\(416\) 1.43191e9 0.975189
\(417\) −2.43303e8 −0.164313
\(418\) 8.24572e8 0.552219
\(419\) −1.40480e9 −0.932962 −0.466481 0.884531i \(-0.654478\pi\)
−0.466481 + 0.884531i \(0.654478\pi\)
\(420\) 3.51863e8 0.231740
\(421\) −8.16663e8 −0.533403 −0.266702 0.963779i \(-0.585934\pi\)
−0.266702 + 0.963779i \(0.585934\pi\)
\(422\) 1.10785e8 0.0717609
\(423\) 6.89875e8 0.443179
\(424\) −7.34490e8 −0.467956
\(425\) 3.06815e8 0.193872
\(426\) −5.37600e7 −0.0336919
\(427\) 7.05070e8 0.438263
\(428\) −2.09528e9 −1.29178
\(429\) −9.21008e8 −0.563200
\(430\) −2.36347e8 −0.143354
\(431\) 2.46225e9 1.48137 0.740683 0.671855i \(-0.234502\pi\)
0.740683 + 0.671855i \(0.234502\pi\)
\(432\) 2.17382e8 0.129727
\(433\) −1.20496e9 −0.713289 −0.356645 0.934240i \(-0.616079\pi\)
−0.356645 + 0.934240i \(0.616079\pi\)
\(434\) −2.64908e8 −0.155554
\(435\) −5.39846e7 −0.0314454
\(436\) 1.34687e9 0.778254
\(437\) −5.51843e9 −3.16323
\(438\) −2.16704e8 −0.123228
\(439\) 7.43862e8 0.419630 0.209815 0.977741i \(-0.432714\pi\)
0.209815 + 0.977741i \(0.432714\pi\)
\(440\) −3.06598e8 −0.171587
\(441\) 6.43151e8 0.357090
\(442\) −1.48709e8 −0.0819143
\(443\) 5.65900e8 0.309262 0.154631 0.987972i \(-0.450581\pi\)
0.154631 + 0.987972i \(0.450581\pi\)
\(444\) −9.80413e8 −0.531580
\(445\) −5.69487e8 −0.306354
\(446\) −9.84510e8 −0.525470
\(447\) −4.69484e8 −0.248625
\(448\) −1.05439e9 −0.554022
\(449\) −1.60627e9 −0.837443 −0.418722 0.908115i \(-0.637522\pi\)
−0.418722 + 0.908115i \(0.637522\pi\)
\(450\) −1.95096e8 −0.100926
\(451\) −1.81005e9 −0.929121
\(452\) −1.78838e9 −0.910909
\(453\) 3.96015e8 0.200156
\(454\) −1.00897e9 −0.506036
\(455\) −1.03026e9 −0.512750
\(456\) 1.41522e9 0.698954
\(457\) −1.42935e9 −0.700538 −0.350269 0.936649i \(-0.613910\pi\)
−0.350269 + 0.936649i \(0.613910\pi\)
\(458\) 6.97277e8 0.339138
\(459\) −8.57779e7 −0.0414029
\(460\) 9.64579e8 0.462046
\(461\) 6.78035e8 0.322328 0.161164 0.986928i \(-0.448475\pi\)
0.161164 + 0.986928i \(0.448475\pi\)
\(462\) −5.09364e8 −0.240315
\(463\) 7.42705e7 0.0347763 0.0173881 0.999849i \(-0.494465\pi\)
0.0173881 + 0.999849i \(0.494465\pi\)
\(464\) 2.51292e8 0.116779
\(465\) 1.26598e8 0.0583905
\(466\) 1.19523e9 0.547143
\(467\) −8.65991e8 −0.393464 −0.196732 0.980457i \(-0.563033\pi\)
−0.196732 + 0.980457i \(0.563033\pi\)
\(468\) −7.43090e8 −0.335105
\(469\) 3.79357e9 1.69802
\(470\) −3.16104e8 −0.140439
\(471\) 1.11051e9 0.489721
\(472\) −1.88578e8 −0.0825458
\(473\) −2.68867e9 −1.16821
\(474\) −3.90239e8 −0.168309
\(475\) 4.01900e9 1.72064
\(476\) 6.46301e8 0.274669
\(477\) 5.83146e8 0.246016
\(478\) −1.30156e8 −0.0545088
\(479\) 3.87827e9 1.61237 0.806183 0.591666i \(-0.201530\pi\)
0.806183 + 0.591666i \(0.201530\pi\)
\(480\) −3.78454e8 −0.156196
\(481\) 2.87066e9 1.17618
\(482\) −7.56177e8 −0.307581
\(483\) 3.40891e9 1.37658
\(484\) 5.73176e8 0.229789
\(485\) −2.49803e8 −0.0994266
\(486\) 5.45440e7 0.0215536
\(487\) 3.18862e9 1.25098 0.625492 0.780231i \(-0.284898\pi\)
0.625492 + 0.780231i \(0.284898\pi\)
\(488\) −4.95686e8 −0.193080
\(489\) −2.76115e7 −0.0106785
\(490\) −2.94695e8 −0.113158
\(491\) 1.59045e9 0.606364 0.303182 0.952933i \(-0.401951\pi\)
0.303182 + 0.952933i \(0.401951\pi\)
\(492\) −1.46039e9 −0.552828
\(493\) −9.91588e7 −0.0372707
\(494\) −1.94796e9 −0.727000
\(495\) 2.43423e8 0.0902076
\(496\) −5.89300e8 −0.216846
\(497\) −6.84114e8 −0.249966
\(498\) 1.93856e8 0.0703359
\(499\) −3.98538e9 −1.43588 −0.717940 0.696105i \(-0.754915\pi\)
−0.717940 + 0.696105i \(0.754915\pi\)
\(500\) −1.48203e9 −0.530227
\(501\) −1.22603e8 −0.0435580
\(502\) −1.54682e9 −0.545729
\(503\) 4.29501e9 1.50479 0.752396 0.658711i \(-0.228898\pi\)
0.752396 + 0.658711i \(0.228898\pi\)
\(504\) −8.74229e8 −0.304171
\(505\) −1.24016e8 −0.0428507
\(506\) −1.39635e9 −0.479144
\(507\) 4.81565e8 0.164107
\(508\) 1.98535e9 0.671914
\(509\) −2.62335e9 −0.881746 −0.440873 0.897570i \(-0.645331\pi\)
−0.440873 + 0.897570i \(0.645331\pi\)
\(510\) 3.93039e7 0.0131202
\(511\) −2.75764e9 −0.914248
\(512\) 3.05967e9 1.00746
\(513\) −1.12361e9 −0.367457
\(514\) 3.97013e8 0.128954
\(515\) −9.31213e8 −0.300417
\(516\) −2.16927e9 −0.695089
\(517\) −3.59598e9 −1.14446
\(518\) 1.58762e9 0.501871
\(519\) 3.93237e8 0.123472
\(520\) 7.24303e8 0.225896
\(521\) −4.65644e9 −1.44252 −0.721260 0.692664i \(-0.756437\pi\)
−0.721260 + 0.692664i \(0.756437\pi\)
\(522\) 6.30526e7 0.0194024
\(523\) −5.16451e9 −1.57860 −0.789301 0.614006i \(-0.789557\pi\)
−0.789301 + 0.614006i \(0.789557\pi\)
\(524\) −4.36026e9 −1.32389
\(525\) −2.48266e9 −0.748790
\(526\) 8.38151e8 0.251115
\(527\) 2.32535e8 0.0692073
\(528\) −1.13311e9 −0.335005
\(529\) 5.94019e9 1.74464
\(530\) −2.67200e8 −0.0779599
\(531\) 1.49721e8 0.0433963
\(532\) 8.46595e9 2.43773
\(533\) 4.27603e9 1.22319
\(534\) 6.65145e8 0.189026
\(535\) −1.62148e9 −0.457799
\(536\) −2.66700e9 −0.748076
\(537\) −2.05124e9 −0.571618
\(538\) 1.34525e9 0.372447
\(539\) −3.35243e9 −0.922146
\(540\) 1.96399e8 0.0536736
\(541\) 7.10793e9 1.92998 0.964990 0.262288i \(-0.0844772\pi\)
0.964990 + 0.262288i \(0.0844772\pi\)
\(542\) −1.84948e9 −0.498944
\(543\) −2.91768e9 −0.782057
\(544\) −6.95144e8 −0.185131
\(545\) 1.04230e9 0.275808
\(546\) 1.20331e9 0.316377
\(547\) 4.92835e9 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(548\) 5.42275e9 1.40763
\(549\) 3.93549e8 0.101507
\(550\) 1.01694e9 0.260631
\(551\) −1.29889e9 −0.330782
\(552\) −2.39657e9 −0.606461
\(553\) −4.96592e9 −1.24871
\(554\) −1.61355e9 −0.403179
\(555\) −7.58717e8 −0.188388
\(556\) 1.02323e9 0.252470
\(557\) −5.04922e9 −1.23803 −0.619015 0.785379i \(-0.712468\pi\)
−0.619015 + 0.785379i \(0.712468\pi\)
\(558\) −1.47863e8 −0.0360281
\(559\) 6.35166e9 1.53796
\(560\) −1.26751e9 −0.304996
\(561\) 4.47119e8 0.106918
\(562\) 2.22478e9 0.528701
\(563\) −1.36397e9 −0.322125 −0.161063 0.986944i \(-0.551492\pi\)
−0.161063 + 0.986944i \(0.551492\pi\)
\(564\) −2.90132e9 −0.680955
\(565\) −1.38398e9 −0.322820
\(566\) 4.69566e8 0.108853
\(567\) 6.94091e8 0.159910
\(568\) 4.80953e8 0.110124
\(569\) −1.81850e9 −0.413829 −0.206915 0.978359i \(-0.566342\pi\)
−0.206915 + 0.978359i \(0.566342\pi\)
\(570\) 5.14845e8 0.116443
\(571\) −9.20531e8 −0.206925 −0.103462 0.994633i \(-0.532992\pi\)
−0.103462 + 0.994633i \(0.532992\pi\)
\(572\) 3.87336e9 0.865371
\(573\) −3.57721e9 −0.794334
\(574\) 2.36486e9 0.521932
\(575\) −6.80585e9 −1.49295
\(576\) −5.88526e8 −0.128318
\(577\) −6.37038e9 −1.38054 −0.690271 0.723551i \(-0.742509\pi\)
−0.690271 + 0.723551i \(0.742509\pi\)
\(578\) −1.48761e9 −0.320437
\(579\) 2.72991e9 0.584484
\(580\) 2.27036e8 0.0483167
\(581\) 2.46689e9 0.521834
\(582\) 2.91764e8 0.0613481
\(583\) −3.03966e9 −0.635308
\(584\) 1.93871e9 0.402779
\(585\) −5.75058e8 −0.118759
\(586\) −1.27265e9 −0.261256
\(587\) 4.67433e9 0.953863 0.476931 0.878941i \(-0.341749\pi\)
0.476931 + 0.878941i \(0.341749\pi\)
\(588\) −2.70482e9 −0.548678
\(589\) 3.04600e9 0.614224
\(590\) −6.86030e7 −0.0137519
\(591\) 2.70744e8 0.0539513
\(592\) 3.53174e9 0.699620
\(593\) 5.82162e9 1.14644 0.573221 0.819401i \(-0.305694\pi\)
0.573221 + 0.819401i \(0.305694\pi\)
\(594\) −2.84311e8 −0.0556598
\(595\) 5.00156e8 0.0973410
\(596\) 1.97445e9 0.382018
\(597\) 1.71484e9 0.329847
\(598\) 3.29870e9 0.630796
\(599\) 6.17800e9 1.17450 0.587251 0.809405i \(-0.300210\pi\)
0.587251 + 0.809405i \(0.300210\pi\)
\(600\) 1.74539e9 0.329885
\(601\) 8.32412e9 1.56415 0.782074 0.623186i \(-0.214162\pi\)
0.782074 + 0.623186i \(0.214162\pi\)
\(602\) 3.51279e9 0.656242
\(603\) 2.11746e9 0.393282
\(604\) −1.66547e9 −0.307544
\(605\) 4.43566e8 0.0814356
\(606\) 1.44847e8 0.0264397
\(607\) −6.60638e9 −1.19896 −0.599478 0.800391i \(-0.704625\pi\)
−0.599478 + 0.800391i \(0.704625\pi\)
\(608\) −9.10576e9 −1.64306
\(609\) 8.02366e8 0.143950
\(610\) −1.80326e8 −0.0321665
\(611\) 8.49509e9 1.50669
\(612\) 3.60745e8 0.0636166
\(613\) −5.00900e9 −0.878293 −0.439146 0.898415i \(-0.644719\pi\)
−0.439146 + 0.898415i \(0.644719\pi\)
\(614\) 2.10860e9 0.367625
\(615\) −1.13016e9 −0.195919
\(616\) 4.55693e9 0.785488
\(617\) 8.30278e9 1.42307 0.711534 0.702652i \(-0.248001\pi\)
0.711534 + 0.702652i \(0.248001\pi\)
\(618\) 1.08763e9 0.185363
\(619\) −4.13223e9 −0.700272 −0.350136 0.936699i \(-0.613865\pi\)
−0.350136 + 0.936699i \(0.613865\pi\)
\(620\) −5.32418e8 −0.0897184
\(621\) 1.90275e9 0.318831
\(622\) −1.52641e9 −0.254333
\(623\) 8.46420e9 1.40242
\(624\) 2.67683e9 0.441037
\(625\) 4.35335e9 0.713252
\(626\) 2.80082e9 0.456325
\(627\) 5.85685e9 0.948916
\(628\) −4.67032e9 −0.752468
\(629\) −1.39361e9 −0.223287
\(630\) −3.18036e8 −0.0506739
\(631\) 8.48866e8 0.134504 0.0672522 0.997736i \(-0.478577\pi\)
0.0672522 + 0.997736i \(0.478577\pi\)
\(632\) 3.49120e9 0.550129
\(633\) 7.86895e8 0.123312
\(634\) −9.60948e7 −0.0149757
\(635\) 1.53641e9 0.238122
\(636\) −2.45246e9 −0.378009
\(637\) 7.91974e9 1.21401
\(638\) −3.28662e8 −0.0501046
\(639\) −3.81852e8 −0.0578951
\(640\) 2.06382e9 0.311202
\(641\) 4.40906e9 0.661216 0.330608 0.943768i \(-0.392746\pi\)
0.330608 + 0.943768i \(0.392746\pi\)
\(642\) 1.89385e9 0.282470
\(643\) −1.11714e10 −1.65717 −0.828586 0.559862i \(-0.810854\pi\)
−0.828586 + 0.559862i \(0.810854\pi\)
\(644\) −1.43364e10 −2.11514
\(645\) −1.67875e9 −0.246335
\(646\) 9.45667e8 0.138014
\(647\) −1.09926e10 −1.59564 −0.797818 0.602899i \(-0.794012\pi\)
−0.797818 + 0.602899i \(0.794012\pi\)
\(648\) −4.87968e8 −0.0704496
\(649\) −7.80424e8 −0.112066
\(650\) −2.40240e9 −0.343122
\(651\) −1.88161e9 −0.267299
\(652\) 1.16122e8 0.0164077
\(653\) −5.52901e9 −0.777053 −0.388527 0.921437i \(-0.627016\pi\)
−0.388527 + 0.921437i \(0.627016\pi\)
\(654\) −1.21738e9 −0.170179
\(655\) −3.37430e9 −0.469179
\(656\) 5.26075e9 0.727585
\(657\) −1.53923e9 −0.211751
\(658\) 4.69821e9 0.642898
\(659\) −8.17686e9 −1.11298 −0.556491 0.830854i \(-0.687852\pi\)
−0.556491 + 0.830854i \(0.687852\pi\)
\(660\) −1.02373e9 −0.138606
\(661\) 1.00210e10 1.34960 0.674801 0.737999i \(-0.264229\pi\)
0.674801 + 0.737999i \(0.264229\pi\)
\(662\) 2.64694e9 0.354603
\(663\) −1.05627e9 −0.140759
\(664\) −1.73430e9 −0.229898
\(665\) 6.55158e9 0.863914
\(666\) 8.86161e8 0.116239
\(667\) 2.19957e9 0.287010
\(668\) 5.15614e8 0.0669279
\(669\) −6.99287e9 −0.902951
\(670\) −9.70229e8 −0.124627
\(671\) −2.05138e9 −0.262130
\(672\) 5.62491e9 0.715029
\(673\) −1.41401e10 −1.78814 −0.894068 0.447931i \(-0.852161\pi\)
−0.894068 + 0.447931i \(0.852161\pi\)
\(674\) −2.78443e9 −0.350290
\(675\) −1.38575e9 −0.173429
\(676\) −2.02526e9 −0.252154
\(677\) 9.91668e9 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(678\) 1.61645e9 0.199186
\(679\) 3.71280e9 0.455152
\(680\) −3.51625e8 −0.0428843
\(681\) −7.16659e9 −0.869556
\(682\) 7.70740e8 0.0930384
\(683\) −1.13006e10 −1.35715 −0.678575 0.734531i \(-0.737402\pi\)
−0.678575 + 0.734531i \(0.737402\pi\)
\(684\) 4.72543e9 0.564606
\(685\) 4.19653e9 0.498854
\(686\) 2.91400e8 0.0344632
\(687\) 4.95269e9 0.582764
\(688\) 7.81437e9 0.914817
\(689\) 7.18083e9 0.836388
\(690\) −8.71849e8 −0.101034
\(691\) 6.57667e9 0.758286 0.379143 0.925338i \(-0.376219\pi\)
0.379143 + 0.925338i \(0.376219\pi\)
\(692\) −1.65379e9 −0.189718
\(693\) −3.61796e9 −0.412950
\(694\) −2.50193e9 −0.284130
\(695\) 7.91849e8 0.0894737
\(696\) −5.64088e8 −0.0634183
\(697\) −2.07587e9 −0.232212
\(698\) −1.45314e8 −0.0161738
\(699\) 8.48960e9 0.940194
\(700\) 1.04410e10 1.15053
\(701\) −1.12915e10 −1.23805 −0.619027 0.785370i \(-0.712473\pi\)
−0.619027 + 0.785370i \(0.712473\pi\)
\(702\) 6.71653e8 0.0732765
\(703\) −1.82550e10 −1.98170
\(704\) 3.06770e9 0.331366
\(705\) −2.24526e9 −0.241326
\(706\) 3.25530e9 0.348157
\(707\) 1.84323e9 0.196161
\(708\) −6.29663e8 −0.0666794
\(709\) 4.04329e9 0.426062 0.213031 0.977045i \(-0.431666\pi\)
0.213031 + 0.977045i \(0.431666\pi\)
\(710\) 1.74966e8 0.0183464
\(711\) −2.77182e9 −0.289216
\(712\) −5.95060e9 −0.617846
\(713\) −5.15816e9 −0.532944
\(714\) −5.84168e8 −0.0600612
\(715\) 2.99750e9 0.306682
\(716\) 8.62662e9 0.878305
\(717\) −9.24485e8 −0.0936662
\(718\) 5.72825e9 0.577546
\(719\) −1.82139e10 −1.82748 −0.913739 0.406302i \(-0.866818\pi\)
−0.913739 + 0.406302i \(0.866818\pi\)
\(720\) −7.07487e8 −0.0706407
\(721\) 1.38405e10 1.37524
\(722\) 8.98954e9 0.888908
\(723\) −5.37105e9 −0.528537
\(724\) 1.22705e10 1.20165
\(725\) −1.60192e9 −0.156119
\(726\) −5.18073e8 −0.0502473
\(727\) 8.36470e9 0.807384 0.403692 0.914895i \(-0.367727\pi\)
0.403692 + 0.914895i \(0.367727\pi\)
\(728\) −1.07652e10 −1.03410
\(729\) 3.87420e8 0.0370370
\(730\) 7.05283e8 0.0671016
\(731\) −3.08352e9 −0.291968
\(732\) −1.65510e9 −0.155968
\(733\) 1.25663e10 1.17853 0.589267 0.807939i \(-0.299417\pi\)
0.589267 + 0.807939i \(0.299417\pi\)
\(734\) −2.92581e9 −0.273092
\(735\) −2.09319e9 −0.194448
\(736\) 1.54199e10 1.42563
\(737\) −1.10373e10 −1.01561
\(738\) 1.31999e9 0.120885
\(739\) −1.15684e10 −1.05443 −0.527214 0.849733i \(-0.676763\pi\)
−0.527214 + 0.849733i \(0.676763\pi\)
\(740\) 3.19084e9 0.289463
\(741\) −1.38361e10 −1.24925
\(742\) 3.97136e9 0.356883
\(743\) 1.87018e10 1.67272 0.836360 0.548181i \(-0.184679\pi\)
0.836360 + 0.548181i \(0.184679\pi\)
\(744\) 1.32283e9 0.117760
\(745\) 1.52797e9 0.135385
\(746\) −3.30699e9 −0.291640
\(747\) 1.37694e9 0.120863
\(748\) −1.88039e9 −0.164283
\(749\) 2.40999e10 2.09570
\(750\) 1.33955e9 0.115943
\(751\) −1.25529e10 −1.08145 −0.540723 0.841201i \(-0.681849\pi\)
−0.540723 + 0.841201i \(0.681849\pi\)
\(752\) 1.04514e10 0.896215
\(753\) −1.09869e10 −0.937763
\(754\) 7.76427e8 0.0659631
\(755\) −1.28886e9 −0.108992
\(756\) −2.91905e9 −0.245706
\(757\) −9.74904e9 −0.816820 −0.408410 0.912799i \(-0.633917\pi\)
−0.408410 + 0.912799i \(0.633917\pi\)
\(758\) −2.61115e9 −0.217766
\(759\) −9.91810e9 −0.823346
\(760\) −4.60597e9 −0.380604
\(761\) 1.96148e10 1.61338 0.806692 0.590972i \(-0.201255\pi\)
0.806692 + 0.590972i \(0.201255\pi\)
\(762\) −1.79449e9 −0.146926
\(763\) −1.54916e10 −1.26259
\(764\) 1.50442e10 1.22051
\(765\) 2.79171e8 0.0225453
\(766\) 2.63339e9 0.211697
\(767\) 1.84366e9 0.147536
\(768\) 3.79565e8 0.0302358
\(769\) 1.59863e9 0.126767 0.0633834 0.997989i \(-0.479811\pi\)
0.0633834 + 0.997989i \(0.479811\pi\)
\(770\) 1.65777e9 0.130860
\(771\) 2.81994e9 0.221590
\(772\) −1.14808e10 −0.898074
\(773\) −2.19525e10 −1.70945 −0.854725 0.519081i \(-0.826274\pi\)
−0.854725 + 0.519081i \(0.826274\pi\)
\(774\) 1.96073e9 0.151993
\(775\) 3.75662e9 0.289895
\(776\) −2.61021e9 −0.200521
\(777\) 1.12767e10 0.862400
\(778\) −6.85795e7 −0.00522115
\(779\) −2.71920e10 −2.06091
\(780\) 2.41845e9 0.182476
\(781\) 1.99041e9 0.149508
\(782\) −1.60141e9 −0.119751
\(783\) 4.47856e8 0.0333405
\(784\) 9.74356e9 0.722123
\(785\) −3.61424e9 −0.266670
\(786\) 3.94109e9 0.289492
\(787\) 2.94002e9 0.215000 0.107500 0.994205i \(-0.465715\pi\)
0.107500 + 0.994205i \(0.465715\pi\)
\(788\) −1.13863e9 −0.0828975
\(789\) 5.95330e9 0.431507
\(790\) 1.27006e9 0.0916496
\(791\) 2.05699e10 1.47780
\(792\) 2.54354e9 0.181928
\(793\) 4.84614e9 0.345096
\(794\) 2.45984e9 0.174395
\(795\) −1.89790e9 −0.133964
\(796\) −7.21186e9 −0.506818
\(797\) −2.52071e10 −1.76367 −0.881836 0.471556i \(-0.843693\pi\)
−0.881836 + 0.471556i \(0.843693\pi\)
\(798\) −7.65207e9 −0.533051
\(799\) −4.12408e9 −0.286031
\(800\) −1.12301e10 −0.775475
\(801\) 4.72446e9 0.324817
\(802\) −9.03969e8 −0.0618790
\(803\) 8.02325e9 0.546822
\(804\) −8.90511e9 −0.604287
\(805\) −1.10946e10 −0.749592
\(806\) −1.82078e9 −0.122486
\(807\) 9.55517e9 0.640002
\(808\) −1.29585e9 −0.0864201
\(809\) −4.67981e9 −0.310748 −0.155374 0.987856i \(-0.549658\pi\)
−0.155374 + 0.987856i \(0.549658\pi\)
\(810\) −1.77518e8 −0.0117367
\(811\) 1.13390e10 0.746451 0.373225 0.927741i \(-0.378252\pi\)
0.373225 + 0.927741i \(0.378252\pi\)
\(812\) −3.37441e9 −0.221183
\(813\) −1.31367e10 −0.857370
\(814\) −4.61912e9 −0.300175
\(815\) 8.98639e7 0.00581479
\(816\) −1.29951e9 −0.0837268
\(817\) −4.03913e10 −2.59126
\(818\) 4.45820e9 0.284789
\(819\) 8.54701e9 0.543652
\(820\) 4.75295e9 0.301033
\(821\) −2.40573e10 −1.51721 −0.758606 0.651549i \(-0.774119\pi\)
−0.758606 + 0.651549i \(0.774119\pi\)
\(822\) −4.90143e9 −0.307802
\(823\) 3.05338e10 1.90933 0.954666 0.297681i \(-0.0962131\pi\)
0.954666 + 0.297681i \(0.0962131\pi\)
\(824\) −9.73030e9 −0.605872
\(825\) 7.22322e9 0.447860
\(826\) 1.01964e9 0.0629529
\(827\) −2.87404e10 −1.76695 −0.883473 0.468482i \(-0.844801\pi\)
−0.883473 + 0.468482i \(0.844801\pi\)
\(828\) −8.00214e9 −0.489892
\(829\) −1.64553e10 −1.00315 −0.501575 0.865114i \(-0.667246\pi\)
−0.501575 + 0.865114i \(0.667246\pi\)
\(830\) −6.30921e8 −0.0383003
\(831\) −1.14609e10 −0.692811
\(832\) −7.24709e9 −0.436246
\(833\) −3.84477e9 −0.230469
\(834\) −9.24859e8 −0.0552070
\(835\) 3.99021e8 0.0237188
\(836\) −2.46314e10 −1.45803
\(837\) −1.05026e9 −0.0619094
\(838\) −5.34000e9 −0.313464
\(839\) 1.71860e10 1.00463 0.502316 0.864684i \(-0.332481\pi\)
0.502316 + 0.864684i \(0.332481\pi\)
\(840\) 2.84525e9 0.165632
\(841\) −1.67322e10 −0.969987
\(842\) −3.10435e9 −0.179217
\(843\) 1.58024e10 0.908503
\(844\) −3.30934e9 −0.189471
\(845\) −1.56729e9 −0.0893618
\(846\) 2.62240e9 0.148903
\(847\) −6.59266e9 −0.372794
\(848\) 8.83449e9 0.497503
\(849\) 3.33528e9 0.187049
\(850\) 1.16629e9 0.0651386
\(851\) 3.09134e10 1.71946
\(852\) 1.60590e9 0.0889571
\(853\) −2.96393e9 −0.163511 −0.0817555 0.996652i \(-0.526053\pi\)
−0.0817555 + 0.996652i \(0.526053\pi\)
\(854\) 2.68016e9 0.147251
\(855\) 3.65689e9 0.200093
\(856\) −1.69430e10 −0.923276
\(857\) −7.20996e9 −0.391291 −0.195646 0.980675i \(-0.562680\pi\)
−0.195646 + 0.980675i \(0.562680\pi\)
\(858\) −3.50100e9 −0.189228
\(859\) −2.08163e10 −1.12054 −0.560269 0.828311i \(-0.689302\pi\)
−0.560269 + 0.828311i \(0.689302\pi\)
\(860\) 7.06008e9 0.378499
\(861\) 1.67974e10 0.896871
\(862\) 9.35968e9 0.497720
\(863\) 1.95918e9 0.103761 0.0518807 0.998653i \(-0.483478\pi\)
0.0518807 + 0.998653i \(0.483478\pi\)
\(864\) 3.13965e9 0.165609
\(865\) −1.27982e9 −0.0672347
\(866\) −4.58038e9 −0.239656
\(867\) −1.05664e10 −0.550628
\(868\) 7.91325e9 0.410711
\(869\) 1.44482e10 0.746868
\(870\) −2.05210e8 −0.0105653
\(871\) 2.60743e10 1.33705
\(872\) 1.08911e10 0.556242
\(873\) 2.07237e9 0.105419
\(874\) −2.09770e10 −1.06281
\(875\) 1.70463e10 0.860204
\(876\) 6.47334e9 0.325360
\(877\) −3.94405e10 −1.97444 −0.987220 0.159366i \(-0.949055\pi\)
−0.987220 + 0.159366i \(0.949055\pi\)
\(878\) 2.82762e9 0.140991
\(879\) −9.03948e9 −0.448934
\(880\) 3.68778e9 0.182422
\(881\) 2.19999e10 1.08394 0.541970 0.840398i \(-0.317679\pi\)
0.541970 + 0.840398i \(0.317679\pi\)
\(882\) 2.44479e9 0.119978
\(883\) −8.61343e8 −0.0421031 −0.0210515 0.999778i \(-0.506701\pi\)
−0.0210515 + 0.999778i \(0.506701\pi\)
\(884\) 4.44220e9 0.216279
\(885\) −4.87280e8 −0.0236307
\(886\) 2.15114e9 0.103908
\(887\) 4.62675e9 0.222609 0.111305 0.993786i \(-0.464497\pi\)
0.111305 + 0.993786i \(0.464497\pi\)
\(888\) −7.92787e9 −0.379937
\(889\) −2.28354e10 −1.09007
\(890\) −2.16477e9 −0.102931
\(891\) −2.01943e9 −0.0956440
\(892\) 2.94090e10 1.38741
\(893\) −5.40217e10 −2.53857
\(894\) −1.78463e9 −0.0835348
\(895\) 6.67592e9 0.311265
\(896\) −3.06743e10 −1.42461
\(897\) 2.34304e10 1.08394
\(898\) −6.10584e9 −0.281370
\(899\) −1.21409e9 −0.0557305
\(900\) 5.82785e9 0.266477
\(901\) −3.48605e9 −0.158781
\(902\) −6.88047e9 −0.312173
\(903\) 2.49510e10 1.12767
\(904\) −1.44613e10 −0.651054
\(905\) 9.49583e9 0.425856
\(906\) 1.50536e9 0.0672499
\(907\) −5.22593e9 −0.232562 −0.116281 0.993216i \(-0.537097\pi\)
−0.116281 + 0.993216i \(0.537097\pi\)
\(908\) 3.01396e10 1.33609
\(909\) 1.02884e9 0.0454331
\(910\) −3.91628e9 −0.172278
\(911\) −3.60178e9 −0.157835 −0.0789175 0.996881i \(-0.525146\pi\)
−0.0789175 + 0.996881i \(0.525146\pi\)
\(912\) −1.70224e10 −0.743086
\(913\) −7.17732e9 −0.312115
\(914\) −5.43334e9 −0.235372
\(915\) −1.28084e9 −0.0552739
\(916\) −2.08289e10 −0.895430
\(917\) 5.01517e10 2.14780
\(918\) −3.26065e8 −0.0139109
\(919\) 8.64702e9 0.367504 0.183752 0.982973i \(-0.441176\pi\)
0.183752 + 0.982973i \(0.441176\pi\)
\(920\) 7.79983e9 0.330239
\(921\) 1.49772e10 0.631715
\(922\) 2.57739e9 0.108298
\(923\) −4.70210e9 −0.196828
\(924\) 1.52156e10 0.634507
\(925\) −2.25138e10 −0.935305
\(926\) 2.82322e8 0.0116844
\(927\) 7.72534e9 0.318521
\(928\) 3.62942e9 0.149080
\(929\) −2.75239e9 −0.112630 −0.0563151 0.998413i \(-0.517935\pi\)
−0.0563151 + 0.998413i \(0.517935\pi\)
\(930\) 4.81234e8 0.0196185
\(931\) −5.03630e10 −2.04544
\(932\) −3.57036e10 −1.44463
\(933\) −1.08419e10 −0.437038
\(934\) −3.29186e9 −0.132199
\(935\) −1.45518e9 −0.0582207
\(936\) −6.00881e9 −0.239510
\(937\) 2.00871e10 0.797678 0.398839 0.917021i \(-0.369413\pi\)
0.398839 + 0.917021i \(0.369413\pi\)
\(938\) 1.44204e10 0.570515
\(939\) 1.98939e10 0.784134
\(940\) 9.44258e9 0.370803
\(941\) −5.18639e9 −0.202909 −0.101455 0.994840i \(-0.532350\pi\)
−0.101455 + 0.994840i \(0.532350\pi\)
\(942\) 4.22134e9 0.164540
\(943\) 4.60474e10 1.78819
\(944\) 2.26823e9 0.0877578
\(945\) −2.25898e9 −0.0870764
\(946\) −1.02203e10 −0.392506
\(947\) 3.37384e10 1.29092 0.645460 0.763794i \(-0.276666\pi\)
0.645460 + 0.763794i \(0.276666\pi\)
\(948\) 1.16571e10 0.444387
\(949\) −1.89540e10 −0.719895
\(950\) 1.52773e10 0.578114
\(951\) −6.82551e8 −0.0257338
\(952\) 5.22615e9 0.196315
\(953\) 3.47736e10 1.30144 0.650720 0.759318i \(-0.274467\pi\)
0.650720 + 0.759318i \(0.274467\pi\)
\(954\) 2.21669e9 0.0826582
\(955\) 1.16423e10 0.432542
\(956\) 3.88799e9 0.143920
\(957\) −2.33445e9 −0.0860981
\(958\) 1.47423e10 0.541735
\(959\) −6.23725e10 −2.28364
\(960\) 1.91541e9 0.0698734
\(961\) −2.46655e10 −0.896515
\(962\) 1.09121e10 0.395182
\(963\) 1.34518e10 0.485388
\(964\) 2.25883e10 0.812109
\(965\) −8.88470e9 −0.318271
\(966\) 1.29582e10 0.462513
\(967\) −4.82322e10 −1.71532 −0.857659 0.514219i \(-0.828082\pi\)
−0.857659 + 0.514219i \(0.828082\pi\)
\(968\) 4.63484e9 0.164237
\(969\) 6.71697e9 0.237159
\(970\) −9.49569e8 −0.0334061
\(971\) 6.11947e9 0.214509 0.107255 0.994232i \(-0.465794\pi\)
0.107255 + 0.994232i \(0.465794\pi\)
\(972\) −1.62932e9 −0.0569083
\(973\) −1.17691e10 −0.409591
\(974\) 1.21208e10 0.420315
\(975\) −1.70640e10 −0.589610
\(976\) 5.96215e9 0.205271
\(977\) 4.14878e10 1.42328 0.711638 0.702546i \(-0.247954\pi\)
0.711638 + 0.702546i \(0.247954\pi\)
\(978\) −1.04959e8 −0.00358783
\(979\) −2.46263e10 −0.838803
\(980\) 8.80305e9 0.298774
\(981\) −8.64695e9 −0.292430
\(982\) 6.04571e9 0.203731
\(983\) 4.34606e10 1.45935 0.729674 0.683795i \(-0.239672\pi\)
0.729674 + 0.683795i \(0.239672\pi\)
\(984\) −1.18091e10 −0.395123
\(985\) −8.81158e8 −0.0293783
\(986\) −3.76929e8 −0.0125225
\(987\) 3.33709e10 1.10474
\(988\) 5.81888e10 1.91951
\(989\) 6.83994e10 2.24836
\(990\) 9.25315e8 0.0303086
\(991\) 2.63085e10 0.858694 0.429347 0.903140i \(-0.358744\pi\)
0.429347 + 0.903140i \(0.358744\pi\)
\(992\) −8.51129e9 −0.276824
\(993\) 1.88010e10 0.609338
\(994\) −2.60050e9 −0.0839855
\(995\) −5.58108e9 −0.179613
\(996\) −5.79082e9 −0.185709
\(997\) −4.90773e10 −1.56837 −0.784183 0.620530i \(-0.786918\pi\)
−0.784183 + 0.620530i \(0.786918\pi\)
\(998\) −1.51495e10 −0.482438
\(999\) 6.29431e9 0.199742
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.1.9 18
3.2 odd 2 531.8.a.e.1.10 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.9 18 1.1 even 1 trivial
531.8.a.e.1.10 18 3.2 odd 2