Properties

Label 177.8.a.d.1.7
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 1.7
Root \(-6.66616\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.66616 q^{2} +27.0000 q^{3} -95.8947 q^{4} -15.0142 q^{5} -152.986 q^{6} +499.350 q^{7} +1268.62 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-5.66616 q^{2} +27.0000 q^{3} -95.8947 q^{4} -15.0142 q^{5} -152.986 q^{6} +499.350 q^{7} +1268.62 q^{8} +729.000 q^{9} +85.0729 q^{10} +5251.00 q^{11} -2589.16 q^{12} +114.657 q^{13} -2829.39 q^{14} -405.384 q^{15} +5086.30 q^{16} -16772.2 q^{17} -4130.63 q^{18} +5407.16 q^{19} +1439.78 q^{20} +13482.4 q^{21} -29753.0 q^{22} -74684.5 q^{23} +34252.8 q^{24} -77899.6 q^{25} -649.662 q^{26} +19683.0 q^{27} -47885.0 q^{28} +58710.0 q^{29} +2296.97 q^{30} +194742. q^{31} -191203. q^{32} +141777. q^{33} +95033.9 q^{34} -7497.35 q^{35} -69907.2 q^{36} +319137. q^{37} -30637.8 q^{38} +3095.73 q^{39} -19047.4 q^{40} +172353. q^{41} -76393.7 q^{42} +17400.1 q^{43} -503543. q^{44} -10945.4 q^{45} +423174. q^{46} +701890. q^{47} +137330. q^{48} -574193. q^{49} +441391. q^{50} -452849. q^{51} -10995.0 q^{52} +467762. q^{53} -111527. q^{54} -78839.7 q^{55} +633486. q^{56} +145993. q^{57} -332660. q^{58} +205379. q^{59} +38874.1 q^{60} -878795. q^{61} -1.10344e6 q^{62} +364026. q^{63} +432342. q^{64} -1721.48 q^{65} -803331. q^{66} -301315. q^{67} +1.60836e6 q^{68} -2.01648e6 q^{69} +42481.1 q^{70} +722433. q^{71} +924826. q^{72} +6.53656e6 q^{73} -1.80828e6 q^{74} -2.10329e6 q^{75} -518518. q^{76} +2.62209e6 q^{77} -17540.9 q^{78} +3.78720e6 q^{79} -76366.8 q^{80} +531441. q^{81} -976580. q^{82} -190031. q^{83} -1.29289e6 q^{84} +251821. q^{85} -98591.5 q^{86} +1.58517e6 q^{87} +6.66154e6 q^{88} +2.70581e6 q^{89} +62018.2 q^{90} +57253.7 q^{91} +7.16185e6 q^{92} +5.25803e6 q^{93} -3.97702e6 q^{94} -81184.2 q^{95} -5.16249e6 q^{96} +1.47369e7 q^{97} +3.25347e6 q^{98} +3.82798e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.66616 −0.500822 −0.250411 0.968140i \(-0.580566\pi\)
−0.250411 + 0.968140i \(0.580566\pi\)
\(3\) 27.0000 0.577350
\(4\) −95.8947 −0.749177
\(5\) −15.0142 −0.0537165 −0.0268582 0.999639i \(-0.508550\pi\)
−0.0268582 + 0.999639i \(0.508550\pi\)
\(6\) −152.986 −0.289150
\(7\) 499.350 0.550252 0.275126 0.961408i \(-0.411280\pi\)
0.275126 + 0.961408i \(0.411280\pi\)
\(8\) 1268.62 0.876027
\(9\) 729.000 0.333333
\(10\) 85.0729 0.0269024
\(11\) 5251.00 1.18951 0.594755 0.803907i \(-0.297249\pi\)
0.594755 + 0.803907i \(0.297249\pi\)
\(12\) −2589.16 −0.432538
\(13\) 114.657 0.0144743 0.00723714 0.999974i \(-0.497696\pi\)
0.00723714 + 0.999974i \(0.497696\pi\)
\(14\) −2829.39 −0.275579
\(15\) −405.384 −0.0310132
\(16\) 5086.30 0.310443
\(17\) −16772.2 −0.827977 −0.413989 0.910282i \(-0.635865\pi\)
−0.413989 + 0.910282i \(0.635865\pi\)
\(18\) −4130.63 −0.166941
\(19\) 5407.16 0.180855 0.0904277 0.995903i \(-0.471177\pi\)
0.0904277 + 0.995903i \(0.471177\pi\)
\(20\) 1439.78 0.0402432
\(21\) 13482.4 0.317688
\(22\) −29753.0 −0.595733
\(23\) −74684.5 −1.27992 −0.639961 0.768408i \(-0.721049\pi\)
−0.639961 + 0.768408i \(0.721049\pi\)
\(24\) 34252.8 0.505774
\(25\) −77899.6 −0.997115
\(26\) −649.662 −0.00724905
\(27\) 19683.0 0.192450
\(28\) −47885.0 −0.412236
\(29\) 58710.0 0.447012 0.223506 0.974703i \(-0.428250\pi\)
0.223506 + 0.974703i \(0.428250\pi\)
\(30\) 2296.97 0.0155321
\(31\) 194742. 1.17407 0.587034 0.809562i \(-0.300296\pi\)
0.587034 + 0.809562i \(0.300296\pi\)
\(32\) −191203. −1.03150
\(33\) 141777. 0.686764
\(34\) 95033.9 0.414670
\(35\) −7497.35 −0.0295576
\(36\) −69907.2 −0.249726
\(37\) 319137. 1.03579 0.517895 0.855444i \(-0.326716\pi\)
0.517895 + 0.855444i \(0.326716\pi\)
\(38\) −30637.8 −0.0905764
\(39\) 3095.73 0.00835673
\(40\) −19047.4 −0.0470571
\(41\) 172353. 0.390549 0.195275 0.980749i \(-0.437440\pi\)
0.195275 + 0.980749i \(0.437440\pi\)
\(42\) −76393.7 −0.159105
\(43\) 17400.1 0.0333742 0.0166871 0.999861i \(-0.494688\pi\)
0.0166871 + 0.999861i \(0.494688\pi\)
\(44\) −503543. −0.891153
\(45\) −10945.4 −0.0179055
\(46\) 423174. 0.641013
\(47\) 701890. 0.986113 0.493057 0.869997i \(-0.335880\pi\)
0.493057 + 0.869997i \(0.335880\pi\)
\(48\) 137330. 0.179234
\(49\) −574193. −0.697223
\(50\) 441391. 0.499377
\(51\) −452849. −0.478033
\(52\) −10995.0 −0.0108438
\(53\) 467762. 0.431578 0.215789 0.976440i \(-0.430768\pi\)
0.215789 + 0.976440i \(0.430768\pi\)
\(54\) −111527. −0.0963833
\(55\) −78839.7 −0.0638963
\(56\) 633486. 0.482036
\(57\) 145993. 0.104417
\(58\) −332660. −0.223874
\(59\) 205379. 0.130189
\(60\) 38874.1 0.0232344
\(61\) −878795. −0.495716 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(62\) −1.10344e6 −0.587999
\(63\) 364026. 0.183417
\(64\) 432342. 0.206157
\(65\) −1721.48 −0.000777508 0
\(66\) −803331. −0.343947
\(67\) −301315. −0.122394 −0.0611968 0.998126i \(-0.519492\pi\)
−0.0611968 + 0.998126i \(0.519492\pi\)
\(68\) 1.60836e6 0.620302
\(69\) −2.01648e6 −0.738963
\(70\) 42481.1 0.0148031
\(71\) 722433. 0.239548 0.119774 0.992801i \(-0.461783\pi\)
0.119774 + 0.992801i \(0.461783\pi\)
\(72\) 924826. 0.292009
\(73\) 6.53656e6 1.96661 0.983307 0.181956i \(-0.0582429\pi\)
0.983307 + 0.181956i \(0.0582429\pi\)
\(74\) −1.80828e6 −0.518746
\(75\) −2.10329e6 −0.575684
\(76\) −518518. −0.135493
\(77\) 2.62209e6 0.654530
\(78\) −17540.9 −0.00418524
\(79\) 3.78720e6 0.864217 0.432109 0.901822i \(-0.357770\pi\)
0.432109 + 0.901822i \(0.357770\pi\)
\(80\) −76366.8 −0.0166759
\(81\) 531441. 0.111111
\(82\) −976580. −0.195596
\(83\) −190031. −0.0364797 −0.0182398 0.999834i \(-0.505806\pi\)
−0.0182398 + 0.999834i \(0.505806\pi\)
\(84\) −1.29289e6 −0.238005
\(85\) 251821. 0.0444760
\(86\) −98591.5 −0.0167146
\(87\) 1.58517e6 0.258083
\(88\) 6.66154e6 1.04204
\(89\) 2.70581e6 0.406848 0.203424 0.979091i \(-0.434793\pi\)
0.203424 + 0.979091i \(0.434793\pi\)
\(90\) 62018.2 0.00896747
\(91\) 57253.7 0.00796451
\(92\) 7.16185e6 0.958888
\(93\) 5.25803e6 0.677848
\(94\) −3.97702e6 −0.493867
\(95\) −81184.2 −0.00971492
\(96\) −5.16249e6 −0.595539
\(97\) 1.47369e7 1.63948 0.819738 0.572739i \(-0.194119\pi\)
0.819738 + 0.572739i \(0.194119\pi\)
\(98\) 3.25347e6 0.349185
\(99\) 3.82798e6 0.396503
\(100\) 7.47015e6 0.747015
\(101\) 3.16363e6 0.305535 0.152767 0.988262i \(-0.451181\pi\)
0.152767 + 0.988262i \(0.451181\pi\)
\(102\) 2.56591e6 0.239410
\(103\) −8.88176e6 −0.800883 −0.400441 0.916322i \(-0.631143\pi\)
−0.400441 + 0.916322i \(0.631143\pi\)
\(104\) 145456. 0.0126799
\(105\) −202428. −0.0170651
\(106\) −2.65041e6 −0.216144
\(107\) 1.68081e7 1.32641 0.663203 0.748440i \(-0.269197\pi\)
0.663203 + 0.748440i \(0.269197\pi\)
\(108\) −1.88749e6 −0.144179
\(109\) 6.20279e6 0.458769 0.229385 0.973336i \(-0.426329\pi\)
0.229385 + 0.973336i \(0.426329\pi\)
\(110\) 446718. 0.0320007
\(111\) 8.61670e6 0.598013
\(112\) 2.53984e6 0.170822
\(113\) 1.00745e7 0.656821 0.328410 0.944535i \(-0.393487\pi\)
0.328410 + 0.944535i \(0.393487\pi\)
\(114\) −827221. −0.0522943
\(115\) 1.12133e6 0.0687529
\(116\) −5.62998e6 −0.334891
\(117\) 83584.6 0.00482476
\(118\) −1.16371e6 −0.0652015
\(119\) −8.37519e6 −0.455596
\(120\) −514279. −0.0271684
\(121\) 8.08587e6 0.414933
\(122\) 4.97939e6 0.248266
\(123\) 4.65354e6 0.225484
\(124\) −1.86747e7 −0.879585
\(125\) 2.34259e6 0.107278
\(126\) −2.06263e6 −0.0918595
\(127\) 8.08324e6 0.350165 0.175082 0.984554i \(-0.443981\pi\)
0.175082 + 0.984554i \(0.443981\pi\)
\(128\) 2.20243e7 0.928256
\(129\) 469802. 0.0192686
\(130\) 9754.17 0.000389393 0
\(131\) 4.08601e7 1.58800 0.793999 0.607919i \(-0.207995\pi\)
0.793999 + 0.607919i \(0.207995\pi\)
\(132\) −1.35957e7 −0.514508
\(133\) 2.70006e6 0.0995161
\(134\) 1.70730e6 0.0612974
\(135\) −295525. −0.0103377
\(136\) −2.12776e7 −0.725331
\(137\) 4.34315e7 1.44306 0.721528 0.692385i \(-0.243440\pi\)
0.721528 + 0.692385i \(0.243440\pi\)
\(138\) 1.14257e7 0.370089
\(139\) −3.00038e7 −0.947598 −0.473799 0.880633i \(-0.657118\pi\)
−0.473799 + 0.880633i \(0.657118\pi\)
\(140\) 718955. 0.0221439
\(141\) 1.89510e7 0.569333
\(142\) −4.09342e6 −0.119971
\(143\) 602062. 0.0172173
\(144\) 3.70791e6 0.103481
\(145\) −881485. −0.0240119
\(146\) −3.70372e7 −0.984924
\(147\) −1.55032e7 −0.402542
\(148\) −3.06036e7 −0.775990
\(149\) −2.00795e7 −0.497279 −0.248640 0.968596i \(-0.579983\pi\)
−0.248640 + 0.968596i \(0.579983\pi\)
\(150\) 1.19176e7 0.288316
\(151\) −4.42468e7 −1.04583 −0.522917 0.852384i \(-0.675156\pi\)
−0.522917 + 0.852384i \(0.675156\pi\)
\(152\) 6.85964e6 0.158434
\(153\) −1.22269e7 −0.275992
\(154\) −1.48572e7 −0.327803
\(155\) −2.92390e6 −0.0630668
\(156\) −296864. −0.00626067
\(157\) −5.91607e7 −1.22007 −0.610035 0.792375i \(-0.708844\pi\)
−0.610035 + 0.792375i \(0.708844\pi\)
\(158\) −2.14588e7 −0.432819
\(159\) 1.26296e7 0.249172
\(160\) 2.87077e6 0.0554088
\(161\) −3.72937e7 −0.704279
\(162\) −3.01123e6 −0.0556469
\(163\) −3.45882e7 −0.625564 −0.312782 0.949825i \(-0.601261\pi\)
−0.312782 + 0.949825i \(0.601261\pi\)
\(164\) −1.65277e7 −0.292590
\(165\) −2.12867e6 −0.0368905
\(166\) 1.07674e6 0.0182698
\(167\) −1.07115e8 −1.77969 −0.889844 0.456265i \(-0.849187\pi\)
−0.889844 + 0.456265i \(0.849187\pi\)
\(168\) 1.71041e7 0.278303
\(169\) −6.27354e7 −0.999790
\(170\) −1.42686e6 −0.0222746
\(171\) 3.94182e6 0.0602851
\(172\) −1.66857e6 −0.0250032
\(173\) 2.97790e7 0.437269 0.218634 0.975807i \(-0.429840\pi\)
0.218634 + 0.975807i \(0.429840\pi\)
\(174\) −8.98183e6 −0.129254
\(175\) −3.88991e7 −0.548664
\(176\) 2.67082e7 0.369275
\(177\) 5.54523e6 0.0751646
\(178\) −1.53315e7 −0.203758
\(179\) 1.22880e8 1.60139 0.800693 0.599075i \(-0.204465\pi\)
0.800693 + 0.599075i \(0.204465\pi\)
\(180\) 1.04960e6 0.0134144
\(181\) 7.94319e7 0.995680 0.497840 0.867269i \(-0.334127\pi\)
0.497840 + 0.867269i \(0.334127\pi\)
\(182\) −324409. −0.00398880
\(183\) −2.37275e7 −0.286202
\(184\) −9.47465e7 −1.12125
\(185\) −4.79160e6 −0.0556390
\(186\) −2.97928e7 −0.339482
\(187\) −8.80708e7 −0.984887
\(188\) −6.73075e7 −0.738773
\(189\) 9.82870e6 0.105896
\(190\) 460003. 0.00486545
\(191\) 1.37986e8 1.43291 0.716453 0.697635i \(-0.245764\pi\)
0.716453 + 0.697635i \(0.245764\pi\)
\(192\) 1.16732e7 0.119025
\(193\) 871407. 0.00872510 0.00436255 0.999990i \(-0.498611\pi\)
0.00436255 + 0.999990i \(0.498611\pi\)
\(194\) −8.35016e7 −0.821086
\(195\) −46479.9 −0.000448894 0
\(196\) 5.50620e7 0.522343
\(197\) 1.71832e8 1.60130 0.800648 0.599135i \(-0.204489\pi\)
0.800648 + 0.599135i \(0.204489\pi\)
\(198\) −2.16899e7 −0.198578
\(199\) −9.52117e7 −0.856455 −0.428227 0.903671i \(-0.640862\pi\)
−0.428227 + 0.903671i \(0.640862\pi\)
\(200\) −9.88251e7 −0.873499
\(201\) −8.13550e6 −0.0706639
\(202\) −1.79256e7 −0.153019
\(203\) 2.93168e7 0.245969
\(204\) 4.34258e7 0.358131
\(205\) −2.58775e6 −0.0209789
\(206\) 5.03255e7 0.401100
\(207\) −5.44450e7 −0.426640
\(208\) 583178. 0.00449344
\(209\) 2.83930e7 0.215129
\(210\) 1.14699e6 0.00854658
\(211\) 1.10926e8 0.812917 0.406458 0.913669i \(-0.366764\pi\)
0.406458 + 0.913669i \(0.366764\pi\)
\(212\) −4.48559e7 −0.323328
\(213\) 1.95057e7 0.138303
\(214\) −9.52375e7 −0.664293
\(215\) −261248. −0.00179275
\(216\) 2.49703e7 0.168591
\(217\) 9.72443e7 0.646033
\(218\) −3.51460e7 −0.229762
\(219\) 1.76487e8 1.13542
\(220\) 7.56031e6 0.0478696
\(221\) −1.92304e6 −0.0119844
\(222\) −4.88236e7 −0.299498
\(223\) −1.48718e8 −0.898042 −0.449021 0.893521i \(-0.648227\pi\)
−0.449021 + 0.893521i \(0.648227\pi\)
\(224\) −9.54774e7 −0.567587
\(225\) −5.67888e7 −0.332372
\(226\) −5.70834e7 −0.328951
\(227\) 4.45860e7 0.252993 0.126496 0.991967i \(-0.459627\pi\)
0.126496 + 0.991967i \(0.459627\pi\)
\(228\) −1.40000e7 −0.0782267
\(229\) −9.60064e7 −0.528294 −0.264147 0.964482i \(-0.585090\pi\)
−0.264147 + 0.964482i \(0.585090\pi\)
\(230\) −6.35363e6 −0.0344330
\(231\) 7.07964e7 0.377893
\(232\) 7.44809e7 0.391595
\(233\) −1.49924e8 −0.776470 −0.388235 0.921560i \(-0.626915\pi\)
−0.388235 + 0.921560i \(0.626915\pi\)
\(234\) −473604. −0.00241635
\(235\) −1.05383e7 −0.0529705
\(236\) −1.96947e7 −0.0975345
\(237\) 1.02254e8 0.498956
\(238\) 4.74551e7 0.228173
\(239\) −4.99588e7 −0.236712 −0.118356 0.992971i \(-0.537762\pi\)
−0.118356 + 0.992971i \(0.537762\pi\)
\(240\) −2.06190e6 −0.00962785
\(241\) 3.94054e6 0.0181341 0.00906705 0.999959i \(-0.497114\pi\)
0.00906705 + 0.999959i \(0.497114\pi\)
\(242\) −4.58158e7 −0.207808
\(243\) 1.43489e7 0.0641500
\(244\) 8.42717e7 0.371379
\(245\) 8.62106e6 0.0374524
\(246\) −2.63677e7 −0.112927
\(247\) 619966. 0.00261775
\(248\) 2.47054e8 1.02851
\(249\) −5.13083e6 −0.0210615
\(250\) −1.32735e7 −0.0537272
\(251\) −6.89865e7 −0.275363 −0.137682 0.990477i \(-0.543965\pi\)
−0.137682 + 0.990477i \(0.543965\pi\)
\(252\) −3.49081e7 −0.137412
\(253\) −3.92169e8 −1.52248
\(254\) −4.58009e7 −0.175370
\(255\) 6.79917e6 0.0256783
\(256\) −1.80133e8 −0.671048
\(257\) 6.74827e7 0.247986 0.123993 0.992283i \(-0.460430\pi\)
0.123993 + 0.992283i \(0.460430\pi\)
\(258\) −2.66197e6 −0.00965016
\(259\) 1.59361e8 0.569945
\(260\) 165081. 0.000582491 0
\(261\) 4.27996e7 0.149004
\(262\) −2.31520e8 −0.795305
\(263\) 7.91658e7 0.268344 0.134172 0.990958i \(-0.457162\pi\)
0.134172 + 0.990958i \(0.457162\pi\)
\(264\) 1.79862e8 0.601623
\(265\) −7.02308e6 −0.0231829
\(266\) −1.52990e7 −0.0498399
\(267\) 7.30568e7 0.234894
\(268\) 2.88945e7 0.0916944
\(269\) 2.24443e8 0.703030 0.351515 0.936182i \(-0.385667\pi\)
0.351515 + 0.936182i \(0.385667\pi\)
\(270\) 1.67449e6 0.00517737
\(271\) 5.85302e7 0.178644 0.0893218 0.996003i \(-0.471530\pi\)
0.0893218 + 0.996003i \(0.471530\pi\)
\(272\) −8.53084e7 −0.257040
\(273\) 1.54585e6 0.00459831
\(274\) −2.46090e8 −0.722715
\(275\) −4.09051e8 −1.18608
\(276\) 1.93370e8 0.553614
\(277\) 3.20396e8 0.905749 0.452874 0.891574i \(-0.350399\pi\)
0.452874 + 0.891574i \(0.350399\pi\)
\(278\) 1.70006e8 0.474578
\(279\) 1.41967e8 0.391356
\(280\) −9.51130e6 −0.0258933
\(281\) −2.95974e8 −0.795758 −0.397879 0.917438i \(-0.630254\pi\)
−0.397879 + 0.917438i \(0.630254\pi\)
\(282\) −1.07380e8 −0.285135
\(283\) −2.63828e8 −0.691940 −0.345970 0.938246i \(-0.612450\pi\)
−0.345970 + 0.938246i \(0.612450\pi\)
\(284\) −6.92775e7 −0.179464
\(285\) −2.19197e6 −0.00560891
\(286\) −3.41138e6 −0.00862281
\(287\) 8.60645e7 0.214901
\(288\) −1.39387e8 −0.343835
\(289\) −1.29032e8 −0.314453
\(290\) 4.99463e6 0.0120257
\(291\) 3.97896e8 0.946552
\(292\) −6.26821e8 −1.47334
\(293\) 3.79473e8 0.881342 0.440671 0.897669i \(-0.354741\pi\)
0.440671 + 0.897669i \(0.354741\pi\)
\(294\) 8.78436e7 0.201602
\(295\) −3.08360e6 −0.00699329
\(296\) 4.04865e8 0.907379
\(297\) 1.03356e8 0.228921
\(298\) 1.13773e8 0.249048
\(299\) −8.56307e6 −0.0185260
\(300\) 2.01694e8 0.431289
\(301\) 8.68872e6 0.0183642
\(302\) 2.50709e8 0.523777
\(303\) 8.54180e7 0.176401
\(304\) 2.75024e7 0.0561453
\(305\) 1.31944e7 0.0266281
\(306\) 6.92797e7 0.138223
\(307\) 4.67759e8 0.922652 0.461326 0.887231i \(-0.347374\pi\)
0.461326 + 0.887231i \(0.347374\pi\)
\(308\) −2.51444e8 −0.490359
\(309\) −2.39808e8 −0.462390
\(310\) 1.65673e7 0.0315853
\(311\) −3.46804e7 −0.0653767 −0.0326883 0.999466i \(-0.510407\pi\)
−0.0326883 + 0.999466i \(0.510407\pi\)
\(312\) 3.92731e6 0.00732072
\(313\) 3.26866e8 0.602511 0.301256 0.953543i \(-0.402594\pi\)
0.301256 + 0.953543i \(0.402594\pi\)
\(314\) 3.35214e8 0.611038
\(315\) −5.46557e6 −0.00985254
\(316\) −3.63172e8 −0.647452
\(317\) 4.00162e8 0.705552 0.352776 0.935708i \(-0.385238\pi\)
0.352776 + 0.935708i \(0.385238\pi\)
\(318\) −7.15612e7 −0.124791
\(319\) 3.08287e8 0.531725
\(320\) −6.49128e6 −0.0110740
\(321\) 4.53819e8 0.765800
\(322\) 2.11312e8 0.352719
\(323\) −9.06899e7 −0.149744
\(324\) −5.09624e7 −0.0832419
\(325\) −8.93170e6 −0.0144325
\(326\) 1.95982e8 0.313296
\(327\) 1.67475e8 0.264871
\(328\) 2.18651e8 0.342132
\(329\) 3.50489e8 0.542611
\(330\) 1.20614e7 0.0184756
\(331\) −2.08461e8 −0.315957 −0.157978 0.987443i \(-0.550498\pi\)
−0.157978 + 0.987443i \(0.550498\pi\)
\(332\) 1.82229e7 0.0273297
\(333\) 2.32651e8 0.345263
\(334\) 6.06932e8 0.891308
\(335\) 4.52400e6 0.00657455
\(336\) 6.85758e7 0.0986242
\(337\) −1.17533e9 −1.67284 −0.836422 0.548086i \(-0.815357\pi\)
−0.836422 + 0.548086i \(0.815357\pi\)
\(338\) 3.55468e8 0.500717
\(339\) 2.72010e8 0.379216
\(340\) −2.41483e7 −0.0333204
\(341\) 1.02259e9 1.39656
\(342\) −2.23350e7 −0.0301921
\(343\) −6.97959e8 −0.933900
\(344\) 2.20741e7 0.0292367
\(345\) 3.02759e7 0.0396945
\(346\) −1.68732e8 −0.218994
\(347\) 3.03478e8 0.389919 0.194960 0.980811i \(-0.437542\pi\)
0.194960 + 0.980811i \(0.437542\pi\)
\(348\) −1.52009e8 −0.193350
\(349\) 7.65904e8 0.964463 0.482232 0.876044i \(-0.339826\pi\)
0.482232 + 0.876044i \(0.339826\pi\)
\(350\) 2.20409e8 0.274783
\(351\) 2.25679e6 0.00278558
\(352\) −1.00401e9 −1.22698
\(353\) 1.07514e9 1.30092 0.650462 0.759539i \(-0.274575\pi\)
0.650462 + 0.759539i \(0.274575\pi\)
\(354\) −3.14202e7 −0.0376441
\(355\) −1.08468e7 −0.0128677
\(356\) −2.59473e8 −0.304801
\(357\) −2.26130e8 −0.263039
\(358\) −6.96258e8 −0.802010
\(359\) −8.01759e8 −0.914562 −0.457281 0.889322i \(-0.651177\pi\)
−0.457281 + 0.889322i \(0.651177\pi\)
\(360\) −1.38855e7 −0.0156857
\(361\) −8.64634e8 −0.967291
\(362\) −4.50074e8 −0.498659
\(363\) 2.18319e8 0.239562
\(364\) −5.49033e6 −0.00596683
\(365\) −9.81413e7 −0.105640
\(366\) 1.34444e8 0.143336
\(367\) 2.39324e8 0.252729 0.126364 0.991984i \(-0.459669\pi\)
0.126364 + 0.991984i \(0.459669\pi\)
\(368\) −3.79868e8 −0.397343
\(369\) 1.25645e8 0.130183
\(370\) 2.71499e7 0.0278652
\(371\) 2.33577e8 0.237477
\(372\) −5.04217e8 −0.507828
\(373\) 1.44976e9 1.44649 0.723245 0.690592i \(-0.242650\pi\)
0.723245 + 0.690592i \(0.242650\pi\)
\(374\) 4.99023e8 0.493253
\(375\) 6.32498e7 0.0619370
\(376\) 8.90434e8 0.863862
\(377\) 6.73149e6 0.00647018
\(378\) −5.56910e7 −0.0530351
\(379\) 5.83143e8 0.550221 0.275111 0.961413i \(-0.411285\pi\)
0.275111 + 0.961413i \(0.411285\pi\)
\(380\) 7.78513e6 0.00727819
\(381\) 2.18247e8 0.202168
\(382\) −7.81850e8 −0.717631
\(383\) −2.12967e9 −1.93694 −0.968472 0.249121i \(-0.919858\pi\)
−0.968472 + 0.249121i \(0.919858\pi\)
\(384\) 5.94657e8 0.535929
\(385\) −3.93686e7 −0.0351591
\(386\) −4.93753e6 −0.00436973
\(387\) 1.26847e7 0.0111247
\(388\) −1.41319e9 −1.22826
\(389\) 1.02632e9 0.884017 0.442008 0.897011i \(-0.354266\pi\)
0.442008 + 0.897011i \(0.354266\pi\)
\(390\) 263363. 0.000224816 0
\(391\) 1.25262e9 1.05975
\(392\) −7.28434e8 −0.610786
\(393\) 1.10322e9 0.916831
\(394\) −9.73626e8 −0.801965
\(395\) −5.68618e7 −0.0464227
\(396\) −3.67083e8 −0.297051
\(397\) 8.48731e8 0.680775 0.340387 0.940285i \(-0.389442\pi\)
0.340387 + 0.940285i \(0.389442\pi\)
\(398\) 5.39484e8 0.428932
\(399\) 7.29017e7 0.0574556
\(400\) −3.96221e8 −0.309547
\(401\) −1.05558e9 −0.817498 −0.408749 0.912647i \(-0.634035\pi\)
−0.408749 + 0.912647i \(0.634035\pi\)
\(402\) 4.60970e7 0.0353901
\(403\) 2.23284e7 0.0169938
\(404\) −3.03375e8 −0.228900
\(405\) −7.97917e6 −0.00596850
\(406\) −1.66114e8 −0.123187
\(407\) 1.67579e9 1.23208
\(408\) −5.74495e8 −0.418770
\(409\) 3.62710e7 0.0262137 0.0131068 0.999914i \(-0.495828\pi\)
0.0131068 + 0.999914i \(0.495828\pi\)
\(410\) 1.46626e7 0.0105067
\(411\) 1.17265e9 0.833149
\(412\) 8.51714e8 0.600003
\(413\) 1.02556e8 0.0716367
\(414\) 3.08494e8 0.213671
\(415\) 2.85316e6 0.00195956
\(416\) −2.19227e7 −0.0149303
\(417\) −8.10102e8 −0.547096
\(418\) −1.60879e8 −0.107742
\(419\) −2.92602e8 −0.194325 −0.0971623 0.995269i \(-0.530977\pi\)
−0.0971623 + 0.995269i \(0.530977\pi\)
\(420\) 1.94118e7 0.0127848
\(421\) −1.22412e9 −0.799534 −0.399767 0.916617i \(-0.630909\pi\)
−0.399767 + 0.916617i \(0.630909\pi\)
\(422\) −6.28526e8 −0.407127
\(423\) 5.11678e8 0.328704
\(424\) 5.93413e8 0.378074
\(425\) 1.30655e9 0.825588
\(426\) −1.10522e8 −0.0692654
\(427\) −4.38826e8 −0.272769
\(428\) −1.61181e9 −0.993712
\(429\) 1.62557e7 0.00994042
\(430\) 1.48027e6 0.000897848 0
\(431\) −6.26512e8 −0.376929 −0.188464 0.982080i \(-0.560351\pi\)
−0.188464 + 0.982080i \(0.560351\pi\)
\(432\) 1.00114e8 0.0597448
\(433\) −2.79784e9 −1.65621 −0.828106 0.560571i \(-0.810582\pi\)
−0.828106 + 0.560571i \(0.810582\pi\)
\(434\) −5.51001e8 −0.323548
\(435\) −2.38001e7 −0.0138633
\(436\) −5.94814e8 −0.343699
\(437\) −4.03831e8 −0.231481
\(438\) −1.00000e9 −0.568646
\(439\) −6.09498e8 −0.343832 −0.171916 0.985112i \(-0.554996\pi\)
−0.171916 + 0.985112i \(0.554996\pi\)
\(440\) −1.00018e8 −0.0559749
\(441\) −4.18587e8 −0.232408
\(442\) 1.08963e7 0.00600205
\(443\) −1.80771e9 −0.987905 −0.493953 0.869489i \(-0.664448\pi\)
−0.493953 + 0.869489i \(0.664448\pi\)
\(444\) −8.26296e8 −0.448018
\(445\) −4.06256e7 −0.0218544
\(446\) 8.42660e8 0.449760
\(447\) −5.42146e8 −0.287104
\(448\) 2.15890e8 0.113438
\(449\) 6.32760e8 0.329896 0.164948 0.986302i \(-0.447254\pi\)
0.164948 + 0.986302i \(0.447254\pi\)
\(450\) 3.21774e8 0.166459
\(451\) 9.05027e8 0.464562
\(452\) −9.66086e8 −0.492075
\(453\) −1.19466e9 −0.603812
\(454\) −2.52631e8 −0.126704
\(455\) −859620. −0.000427825 0
\(456\) 1.85210e8 0.0914720
\(457\) −3.19962e8 −0.156816 −0.0784082 0.996921i \(-0.524984\pi\)
−0.0784082 + 0.996921i \(0.524984\pi\)
\(458\) 5.43987e8 0.264582
\(459\) −3.30127e8 −0.159344
\(460\) −1.07530e8 −0.0515081
\(461\) −9.55318e8 −0.454145 −0.227073 0.973878i \(-0.572915\pi\)
−0.227073 + 0.973878i \(0.572915\pi\)
\(462\) −4.01143e8 −0.189257
\(463\) 3.55134e9 1.66287 0.831437 0.555619i \(-0.187519\pi\)
0.831437 + 0.555619i \(0.187519\pi\)
\(464\) 2.98617e8 0.138772
\(465\) −7.89452e7 −0.0364116
\(466\) 8.49491e8 0.388873
\(467\) 4.08842e9 1.85758 0.928788 0.370613i \(-0.120852\pi\)
0.928788 + 0.370613i \(0.120852\pi\)
\(468\) −8.01532e6 −0.00361460
\(469\) −1.50461e8 −0.0673473
\(470\) 5.97119e7 0.0265288
\(471\) −1.59734e9 −0.704407
\(472\) 2.60548e8 0.114049
\(473\) 9.13679e7 0.0396990
\(474\) −5.79389e8 −0.249888
\(475\) −4.21215e8 −0.180334
\(476\) 8.03136e8 0.341322
\(477\) 3.40999e8 0.143859
\(478\) 2.83075e8 0.118550
\(479\) 2.21317e9 0.920112 0.460056 0.887890i \(-0.347829\pi\)
0.460056 + 0.887890i \(0.347829\pi\)
\(480\) 7.75108e7 0.0319903
\(481\) 3.65912e7 0.0149923
\(482\) −2.23277e7 −0.00908196
\(483\) −1.00693e9 −0.406616
\(484\) −7.75392e8 −0.310858
\(485\) −2.21263e8 −0.0880669
\(486\) −8.13032e7 −0.0321278
\(487\) −4.68349e9 −1.83746 −0.918731 0.394883i \(-0.870785\pi\)
−0.918731 + 0.394883i \(0.870785\pi\)
\(488\) −1.11486e9 −0.434261
\(489\) −9.33882e8 −0.361170
\(490\) −4.88483e7 −0.0187570
\(491\) −1.79902e9 −0.685885 −0.342942 0.939356i \(-0.611424\pi\)
−0.342942 + 0.939356i \(0.611424\pi\)
\(492\) −4.46249e8 −0.168927
\(493\) −9.84696e8 −0.370116
\(494\) −3.51283e6 −0.00131103
\(495\) −5.74742e7 −0.0212988
\(496\) 9.90515e8 0.364481
\(497\) 3.60747e8 0.131812
\(498\) 2.90721e7 0.0105481
\(499\) −8.51942e8 −0.306943 −0.153472 0.988153i \(-0.549045\pi\)
−0.153472 + 0.988153i \(0.549045\pi\)
\(500\) −2.24642e8 −0.0803702
\(501\) −2.89211e9 −1.02750
\(502\) 3.90888e8 0.137908
\(503\) −3.29333e9 −1.15385 −0.576923 0.816798i \(-0.695747\pi\)
−0.576923 + 0.816798i \(0.695747\pi\)
\(504\) 4.61812e8 0.160679
\(505\) −4.74994e7 −0.0164123
\(506\) 2.22209e9 0.762491
\(507\) −1.69386e9 −0.577229
\(508\) −7.75139e8 −0.262335
\(509\) −1.92293e9 −0.646325 −0.323162 0.946344i \(-0.604746\pi\)
−0.323162 + 0.946344i \(0.604746\pi\)
\(510\) −3.85252e7 −0.0128602
\(511\) 3.26403e9 1.08213
\(512\) −1.79845e9 −0.592180
\(513\) 1.06429e8 0.0348056
\(514\) −3.82367e8 −0.124197
\(515\) 1.33353e8 0.0430206
\(516\) −4.50515e7 −0.0144356
\(517\) 3.68563e9 1.17299
\(518\) −9.02965e8 −0.285441
\(519\) 8.04032e8 0.252457
\(520\) −2.18391e6 −0.000681118 0
\(521\) 5.43560e9 1.68390 0.841949 0.539558i \(-0.181409\pi\)
0.841949 + 0.539558i \(0.181409\pi\)
\(522\) −2.42509e8 −0.0746246
\(523\) −3.26406e9 −0.997705 −0.498853 0.866687i \(-0.666245\pi\)
−0.498853 + 0.866687i \(0.666245\pi\)
\(524\) −3.91827e9 −1.18969
\(525\) −1.05028e9 −0.316772
\(526\) −4.48566e8 −0.134393
\(527\) −3.26625e9 −0.972102
\(528\) 7.21121e8 0.213201
\(529\) 2.17296e9 0.638199
\(530\) 3.97939e7 0.0116105
\(531\) 1.49721e8 0.0433963
\(532\) −2.58922e8 −0.0745551
\(533\) 1.97614e7 0.00565292
\(534\) −4.13951e8 −0.117640
\(535\) −2.52361e8 −0.0712498
\(536\) −3.82255e8 −0.107220
\(537\) 3.31776e9 0.924561
\(538\) −1.27173e9 −0.352093
\(539\) −3.01509e9 −0.829353
\(540\) 2.83393e7 0.00774480
\(541\) 6.87940e9 1.86793 0.933964 0.357367i \(-0.116325\pi\)
0.933964 + 0.357367i \(0.116325\pi\)
\(542\) −3.31641e8 −0.0894687
\(543\) 2.14466e9 0.574856
\(544\) 3.20690e9 0.854062
\(545\) −9.31300e7 −0.0246435
\(546\) −8.75903e6 −0.00230294
\(547\) −3.21914e9 −0.840977 −0.420489 0.907298i \(-0.638141\pi\)
−0.420489 + 0.907298i \(0.638141\pi\)
\(548\) −4.16485e9 −1.08110
\(549\) −6.40641e8 −0.165239
\(550\) 2.31775e9 0.594014
\(551\) 3.17454e8 0.0808446
\(552\) −2.55816e9 −0.647351
\(553\) 1.89114e9 0.475537
\(554\) −1.81541e9 −0.453619
\(555\) −1.29373e8 −0.0321232
\(556\) 2.87720e9 0.709918
\(557\) −2.80717e9 −0.688295 −0.344148 0.938916i \(-0.611832\pi\)
−0.344148 + 0.938916i \(0.611832\pi\)
\(558\) −8.04406e8 −0.196000
\(559\) 1.99503e6 0.000483068 0
\(560\) −3.81338e7 −0.00917596
\(561\) −2.37791e9 −0.568625
\(562\) 1.67703e9 0.398533
\(563\) 6.17910e8 0.145930 0.0729652 0.997334i \(-0.476754\pi\)
0.0729652 + 0.997334i \(0.476754\pi\)
\(564\) −1.81730e9 −0.426531
\(565\) −1.51260e8 −0.0352821
\(566\) 1.49489e9 0.346539
\(567\) 2.65375e8 0.0611391
\(568\) 9.16495e8 0.209851
\(569\) −4.23484e9 −0.963705 −0.481852 0.876252i \(-0.660036\pi\)
−0.481852 + 0.876252i \(0.660036\pi\)
\(570\) 1.24201e7 0.00280907
\(571\) 6.61606e8 0.148721 0.0743607 0.997231i \(-0.476308\pi\)
0.0743607 + 0.997231i \(0.476308\pi\)
\(572\) −5.77345e7 −0.0128988
\(573\) 3.72562e9 0.827289
\(574\) −4.87655e8 −0.107627
\(575\) 5.81789e9 1.27623
\(576\) 3.15178e8 0.0687190
\(577\) 9.05424e8 0.196217 0.0981085 0.995176i \(-0.468721\pi\)
0.0981085 + 0.995176i \(0.468721\pi\)
\(578\) 7.31118e8 0.157485
\(579\) 2.35280e7 0.00503744
\(580\) 8.45297e7 0.0179892
\(581\) −9.48919e7 −0.0200730
\(582\) −2.25454e9 −0.474054
\(583\) 2.45622e9 0.513366
\(584\) 8.29242e9 1.72281
\(585\) −1.25496e6 −0.000259169 0
\(586\) −2.15016e9 −0.441396
\(587\) 5.48391e9 1.11907 0.559534 0.828807i \(-0.310980\pi\)
0.559534 + 0.828807i \(0.310980\pi\)
\(588\) 1.48667e9 0.301575
\(589\) 1.05300e9 0.212336
\(590\) 1.74722e7 0.00350240
\(591\) 4.63946e9 0.924509
\(592\) 1.62323e9 0.321554
\(593\) −6.47258e9 −1.27464 −0.637318 0.770601i \(-0.719956\pi\)
−0.637318 + 0.770601i \(0.719956\pi\)
\(594\) −5.85629e8 −0.114649
\(595\) 1.25747e8 0.0244730
\(596\) 1.92551e9 0.372550
\(597\) −2.57072e9 −0.494474
\(598\) 4.85197e7 0.00927821
\(599\) 2.50605e8 0.0476426 0.0238213 0.999716i \(-0.492417\pi\)
0.0238213 + 0.999716i \(0.492417\pi\)
\(600\) −2.66828e9 −0.504315
\(601\) −4.78670e9 −0.899447 −0.449724 0.893168i \(-0.648477\pi\)
−0.449724 + 0.893168i \(0.648477\pi\)
\(602\) −4.92317e7 −0.00919723
\(603\) −2.19658e8 −0.0407978
\(604\) 4.24303e9 0.783514
\(605\) −1.21403e8 −0.0222888
\(606\) −4.83992e8 −0.0883453
\(607\) −2.08136e9 −0.377735 −0.188868 0.982003i \(-0.560482\pi\)
−0.188868 + 0.982003i \(0.560482\pi\)
\(608\) −1.03387e9 −0.186553
\(609\) 7.91555e8 0.142011
\(610\) −7.47616e7 −0.0133360
\(611\) 8.04763e7 0.0142733
\(612\) 1.17250e9 0.206767
\(613\) 9.25255e9 1.62237 0.811185 0.584790i \(-0.198823\pi\)
0.811185 + 0.584790i \(0.198823\pi\)
\(614\) −2.65040e9 −0.462085
\(615\) −6.98692e7 −0.0121122
\(616\) 3.32644e9 0.573386
\(617\) 4.36405e9 0.747982 0.373991 0.927432i \(-0.377989\pi\)
0.373991 + 0.927432i \(0.377989\pi\)
\(618\) 1.35879e9 0.231575
\(619\) −5.18190e9 −0.878156 −0.439078 0.898449i \(-0.644695\pi\)
−0.439078 + 0.898449i \(0.644695\pi\)
\(620\) 2.80386e8 0.0472482
\(621\) −1.47002e9 −0.246321
\(622\) 1.96505e8 0.0327421
\(623\) 1.35114e9 0.223869
\(624\) 1.57458e7 0.00259429
\(625\) 6.05073e9 0.991352
\(626\) −1.85208e9 −0.301751
\(627\) 7.66611e8 0.124205
\(628\) 5.67319e9 0.914048
\(629\) −5.35263e9 −0.857610
\(630\) 3.09688e7 0.00493437
\(631\) 5.15467e9 0.816767 0.408383 0.912810i \(-0.366093\pi\)
0.408383 + 0.912810i \(0.366093\pi\)
\(632\) 4.80452e9 0.757077
\(633\) 2.99501e9 0.469338
\(634\) −2.26738e9 −0.353356
\(635\) −1.21363e8 −0.0188096
\(636\) −1.21111e9 −0.186674
\(637\) −6.58350e7 −0.0100918
\(638\) −1.74680e9 −0.266300
\(639\) 5.26654e8 0.0798495
\(640\) −3.30678e8 −0.0498626
\(641\) −4.91786e9 −0.737518 −0.368759 0.929525i \(-0.620217\pi\)
−0.368759 + 0.929525i \(0.620217\pi\)
\(642\) −2.57141e9 −0.383530
\(643\) −1.42616e9 −0.211558 −0.105779 0.994390i \(-0.533734\pi\)
−0.105779 + 0.994390i \(0.533734\pi\)
\(644\) 3.57627e9 0.527630
\(645\) −7.05371e6 −0.00103504
\(646\) 5.13863e8 0.0749952
\(647\) 6.26814e9 0.909858 0.454929 0.890528i \(-0.349665\pi\)
0.454929 + 0.890528i \(0.349665\pi\)
\(648\) 6.74198e8 0.0973363
\(649\) 1.07845e9 0.154861
\(650\) 5.06084e7 0.00722813
\(651\) 2.62559e9 0.372987
\(652\) 3.31683e9 0.468658
\(653\) 1.00126e10 1.40719 0.703593 0.710603i \(-0.251578\pi\)
0.703593 + 0.710603i \(0.251578\pi\)
\(654\) −9.48941e8 −0.132653
\(655\) −6.13483e8 −0.0853017
\(656\) 8.76640e8 0.121243
\(657\) 4.76515e9 0.655538
\(658\) −1.98592e9 −0.271752
\(659\) 3.69501e9 0.502941 0.251470 0.967865i \(-0.419086\pi\)
0.251470 + 0.967865i \(0.419086\pi\)
\(660\) 2.04128e8 0.0276375
\(661\) 2.49829e9 0.336464 0.168232 0.985747i \(-0.446194\pi\)
0.168232 + 0.985747i \(0.446194\pi\)
\(662\) 1.18117e9 0.158238
\(663\) −5.19221e7 −0.00691919
\(664\) −2.41077e8 −0.0319572
\(665\) −4.05393e7 −0.00534565
\(666\) −1.31824e9 −0.172915
\(667\) −4.38473e9 −0.572140
\(668\) 1.02718e10 1.33330
\(669\) −4.01539e9 −0.518485
\(670\) −2.56337e7 −0.00329268
\(671\) −4.61455e9 −0.589659
\(672\) −2.57789e9 −0.327697
\(673\) 3.99102e9 0.504698 0.252349 0.967636i \(-0.418797\pi\)
0.252349 + 0.967636i \(0.418797\pi\)
\(674\) 6.65961e9 0.837798
\(675\) −1.53330e9 −0.191895
\(676\) 6.01599e9 0.749020
\(677\) −3.69552e9 −0.457737 −0.228868 0.973457i \(-0.573503\pi\)
−0.228868 + 0.973457i \(0.573503\pi\)
\(678\) −1.54125e9 −0.189920
\(679\) 7.35887e9 0.902125
\(680\) 3.19466e8 0.0389622
\(681\) 1.20382e9 0.146065
\(682\) −5.79416e9 −0.699431
\(683\) −6.33943e9 −0.761338 −0.380669 0.924711i \(-0.624306\pi\)
−0.380669 + 0.924711i \(0.624306\pi\)
\(684\) −3.77999e8 −0.0451642
\(685\) −6.52091e8 −0.0775159
\(686\) 3.95475e9 0.467718
\(687\) −2.59217e9 −0.305011
\(688\) 8.85020e7 0.0103608
\(689\) 5.36320e7 0.00624679
\(690\) −1.71548e8 −0.0198799
\(691\) 9.02991e9 1.04114 0.520572 0.853818i \(-0.325719\pi\)
0.520572 + 0.853818i \(0.325719\pi\)
\(692\) −2.85564e9 −0.327592
\(693\) 1.91150e9 0.218177
\(694\) −1.71956e9 −0.195280
\(695\) 4.50483e8 0.0509016
\(696\) 2.01098e9 0.226087
\(697\) −2.89074e9 −0.323366
\(698\) −4.33974e9 −0.483025
\(699\) −4.04794e9 −0.448295
\(700\) 3.73022e9 0.411047
\(701\) −4.09079e9 −0.448533 −0.224267 0.974528i \(-0.571999\pi\)
−0.224267 + 0.974528i \(0.571999\pi\)
\(702\) −1.27873e7 −0.00139508
\(703\) 1.72563e9 0.187328
\(704\) 2.27023e9 0.245226
\(705\) −2.84535e8 −0.0305826
\(706\) −6.09189e9 −0.651532
\(707\) 1.57976e9 0.168121
\(708\) −5.31758e8 −0.0563116
\(709\) −1.70129e10 −1.79273 −0.896366 0.443315i \(-0.853802\pi\)
−0.896366 + 0.443315i \(0.853802\pi\)
\(710\) 6.14595e7 0.00644443
\(711\) 2.76087e9 0.288072
\(712\) 3.43265e9 0.356410
\(713\) −1.45442e10 −1.50271
\(714\) 1.28129e9 0.131736
\(715\) −9.03949e6 −0.000924853 0
\(716\) −1.17835e10 −1.19972
\(717\) −1.34889e9 −0.136666
\(718\) 4.54289e9 0.458033
\(719\) −1.05938e10 −1.06292 −0.531460 0.847083i \(-0.678357\pi\)
−0.531460 + 0.847083i \(0.678357\pi\)
\(720\) −5.56714e7 −0.00555864
\(721\) −4.43511e9 −0.440687
\(722\) 4.89915e9 0.484441
\(723\) 1.06395e8 0.0104697
\(724\) −7.61709e9 −0.745941
\(725\) −4.57349e9 −0.445722
\(726\) −1.23703e9 −0.119978
\(727\) 6.96354e9 0.672140 0.336070 0.941837i \(-0.390902\pi\)
0.336070 + 0.941837i \(0.390902\pi\)
\(728\) 7.26334e7 0.00697712
\(729\) 3.87420e8 0.0370370
\(730\) 5.56084e8 0.0529067
\(731\) −2.91837e8 −0.0276331
\(732\) 2.27534e9 0.214416
\(733\) −2.65861e9 −0.249340 −0.124670 0.992198i \(-0.539787\pi\)
−0.124670 + 0.992198i \(0.539787\pi\)
\(734\) −1.35605e9 −0.126572
\(735\) 2.32768e8 0.0216231
\(736\) 1.42799e10 1.32024
\(737\) −1.58220e9 −0.145588
\(738\) −7.11927e8 −0.0651986
\(739\) −4.87522e8 −0.0444364 −0.0222182 0.999753i \(-0.507073\pi\)
−0.0222182 + 0.999753i \(0.507073\pi\)
\(740\) 4.59488e8 0.0416834
\(741\) 1.67391e7 0.00151136
\(742\) −1.32348e9 −0.118934
\(743\) −1.88059e10 −1.68202 −0.841012 0.541016i \(-0.818040\pi\)
−0.841012 + 0.541016i \(0.818040\pi\)
\(744\) 6.67045e9 0.593813
\(745\) 3.01477e8 0.0267121
\(746\) −8.21457e9 −0.724434
\(747\) −1.38532e8 −0.0121599
\(748\) 8.44552e9 0.737855
\(749\) 8.39314e9 0.729857
\(750\) −3.58384e8 −0.0310194
\(751\) 2.25057e10 1.93889 0.969444 0.245315i \(-0.0788912\pi\)
0.969444 + 0.245315i \(0.0788912\pi\)
\(752\) 3.57003e9 0.306132
\(753\) −1.86264e9 −0.158981
\(754\) −3.81417e7 −0.00324041
\(755\) 6.64331e8 0.0561785
\(756\) −9.42520e8 −0.0793349
\(757\) 9.35131e8 0.0783496 0.0391748 0.999232i \(-0.487527\pi\)
0.0391748 + 0.999232i \(0.487527\pi\)
\(758\) −3.30418e9 −0.275563
\(759\) −1.05886e10 −0.879004
\(760\) −1.02992e8 −0.00851053
\(761\) −8.03138e9 −0.660608 −0.330304 0.943875i \(-0.607151\pi\)
−0.330304 + 0.943875i \(0.607151\pi\)
\(762\) −1.23662e9 −0.101250
\(763\) 3.09736e9 0.252439
\(764\) −1.32321e10 −1.07350
\(765\) 1.83578e8 0.0148253
\(766\) 1.20671e10 0.970065
\(767\) 2.35481e7 0.00188439
\(768\) −4.86359e9 −0.387430
\(769\) −2.58797e9 −0.205219 −0.102610 0.994722i \(-0.532719\pi\)
−0.102610 + 0.994722i \(0.532719\pi\)
\(770\) 2.23069e8 0.0176084
\(771\) 1.82203e9 0.143175
\(772\) −8.35633e7 −0.00653665
\(773\) −1.46523e10 −1.14098 −0.570489 0.821306i \(-0.693246\pi\)
−0.570489 + 0.821306i \(0.693246\pi\)
\(774\) −7.18732e7 −0.00557152
\(775\) −1.51703e10 −1.17068
\(776\) 1.86956e10 1.43622
\(777\) 4.30275e9 0.329058
\(778\) −5.81531e9 −0.442735
\(779\) 9.31941e8 0.0706329
\(780\) 4.45718e6 0.000336301 0
\(781\) 3.79350e9 0.284945
\(782\) −7.09756e9 −0.530744
\(783\) 1.15559e9 0.0860275
\(784\) −2.92052e9 −0.216448
\(785\) 8.88251e8 0.0655378
\(786\) −6.25104e9 −0.459170
\(787\) −7.14096e9 −0.522210 −0.261105 0.965310i \(-0.584087\pi\)
−0.261105 + 0.965310i \(0.584087\pi\)
\(788\) −1.64777e10 −1.19965
\(789\) 2.13748e9 0.154929
\(790\) 3.22188e8 0.0232495
\(791\) 5.03068e9 0.361417
\(792\) 4.85626e9 0.347347
\(793\) −1.00760e8 −0.00717514
\(794\) −4.80904e9 −0.340947
\(795\) −1.89623e8 −0.0133846
\(796\) 9.13029e9 0.641636
\(797\) 1.01959e10 0.713379 0.356689 0.934223i \(-0.383905\pi\)
0.356689 + 0.934223i \(0.383905\pi\)
\(798\) −4.13073e8 −0.0287751
\(799\) −1.17722e10 −0.816480
\(800\) 1.48947e10 1.02853
\(801\) 1.97253e9 0.135616
\(802\) 5.98110e9 0.409421
\(803\) 3.43235e10 2.33931
\(804\) 7.80151e8 0.0529398
\(805\) 5.59936e8 0.0378314
\(806\) −1.26516e8 −0.00851087
\(807\) 6.05997e9 0.405894
\(808\) 4.01345e9 0.267657
\(809\) −1.75531e10 −1.16556 −0.582780 0.812630i \(-0.698035\pi\)
−0.582780 + 0.812630i \(0.698035\pi\)
\(810\) 4.52112e7 0.00298916
\(811\) −2.83520e10 −1.86643 −0.933214 0.359321i \(-0.883008\pi\)
−0.933214 + 0.359321i \(0.883008\pi\)
\(812\) −2.81133e9 −0.184275
\(813\) 1.58032e9 0.103140
\(814\) −9.49529e9 −0.617054
\(815\) 5.19315e8 0.0336031
\(816\) −2.30333e9 −0.148402
\(817\) 9.40849e7 0.00603591
\(818\) −2.05517e8 −0.0131284
\(819\) 4.17380e7 0.00265484
\(820\) 2.48151e8 0.0157169
\(821\) −2.80476e10 −1.76886 −0.884432 0.466669i \(-0.845454\pi\)
−0.884432 + 0.466669i \(0.845454\pi\)
\(822\) −6.64443e9 −0.417260
\(823\) −2.23504e10 −1.39761 −0.698806 0.715311i \(-0.746285\pi\)
−0.698806 + 0.715311i \(0.746285\pi\)
\(824\) −1.12676e10 −0.701595
\(825\) −1.10444e10 −0.684782
\(826\) −5.81098e8 −0.0358773
\(827\) 1.26693e10 0.778900 0.389450 0.921048i \(-0.372665\pi\)
0.389450 + 0.921048i \(0.372665\pi\)
\(828\) 5.22099e9 0.319629
\(829\) −2.22306e10 −1.35522 −0.677610 0.735421i \(-0.736984\pi\)
−0.677610 + 0.735421i \(0.736984\pi\)
\(830\) −1.61665e7 −0.000981391 0
\(831\) 8.65069e9 0.522934
\(832\) 4.95709e7 0.00298397
\(833\) 9.63047e9 0.577285
\(834\) 4.59016e9 0.273998
\(835\) 1.60825e9 0.0955986
\(836\) −2.72274e9 −0.161170
\(837\) 3.83310e9 0.225949
\(838\) 1.65793e9 0.0973221
\(839\) −2.27124e10 −1.32769 −0.663845 0.747870i \(-0.731077\pi\)
−0.663845 + 0.747870i \(0.731077\pi\)
\(840\) −2.56805e8 −0.0149495
\(841\) −1.38030e10 −0.800180
\(842\) 6.93606e9 0.400425
\(843\) −7.99129e9 −0.459431
\(844\) −1.06372e10 −0.609019
\(845\) 9.41922e8 0.0537052
\(846\) −2.89925e9 −0.164622
\(847\) 4.03768e9 0.228318
\(848\) 2.37918e9 0.133981
\(849\) −7.12336e9 −0.399492
\(850\) −7.40310e9 −0.413473
\(851\) −2.38346e10 −1.32573
\(852\) −1.87049e9 −0.103614
\(853\) −2.92944e10 −1.61608 −0.808041 0.589127i \(-0.799472\pi\)
−0.808041 + 0.589127i \(0.799472\pi\)
\(854\) 2.48646e9 0.136609
\(855\) −5.91833e7 −0.00323831
\(856\) 2.13232e10 1.16197
\(857\) −2.01612e9 −0.109417 −0.0547084 0.998502i \(-0.517423\pi\)
−0.0547084 + 0.998502i \(0.517423\pi\)
\(858\) −9.21072e7 −0.00497838
\(859\) −1.62001e10 −0.872052 −0.436026 0.899934i \(-0.643614\pi\)
−0.436026 + 0.899934i \(0.643614\pi\)
\(860\) 2.50523e7 0.00134309
\(861\) 2.32374e9 0.124073
\(862\) 3.54992e9 0.188774
\(863\) 4.82303e9 0.255436 0.127718 0.991811i \(-0.459235\pi\)
0.127718 + 0.991811i \(0.459235\pi\)
\(864\) −3.76346e9 −0.198513
\(865\) −4.47108e8 −0.0234885
\(866\) 1.58530e10 0.829468
\(867\) −3.48387e9 −0.181550
\(868\) −9.32521e9 −0.483993
\(869\) 1.98866e10 1.02799
\(870\) 1.34855e8 0.00694305
\(871\) −3.45477e7 −0.00177156
\(872\) 7.86900e9 0.401894
\(873\) 1.07432e10 0.546492
\(874\) 2.28817e9 0.115931
\(875\) 1.16977e9 0.0590299
\(876\) −1.69242e10 −0.850634
\(877\) 4.98280e9 0.249445 0.124723 0.992192i \(-0.460196\pi\)
0.124723 + 0.992192i \(0.460196\pi\)
\(878\) 3.45351e9 0.172199
\(879\) 1.02458e10 0.508843
\(880\) −4.01003e8 −0.0198362
\(881\) 1.27067e10 0.626061 0.313031 0.949743i \(-0.398656\pi\)
0.313031 + 0.949743i \(0.398656\pi\)
\(882\) 2.37178e9 0.116395
\(883\) −1.01601e9 −0.0496633 −0.0248316 0.999692i \(-0.507905\pi\)
−0.0248316 + 0.999692i \(0.507905\pi\)
\(884\) 1.84409e8 0.00897843
\(885\) −8.32573e7 −0.00403758
\(886\) 1.02428e10 0.494765
\(887\) 1.10442e10 0.531376 0.265688 0.964059i \(-0.414401\pi\)
0.265688 + 0.964059i \(0.414401\pi\)
\(888\) 1.09313e10 0.523876
\(889\) 4.03636e9 0.192679
\(890\) 2.30191e8 0.0109452
\(891\) 2.79060e9 0.132168
\(892\) 1.42613e10 0.672793
\(893\) 3.79523e9 0.178344
\(894\) 3.07188e9 0.143788
\(895\) −1.84495e9 −0.0860208
\(896\) 1.09978e10 0.510775
\(897\) −2.31203e8 −0.0106960
\(898\) −3.58532e9 −0.165219
\(899\) 1.14333e10 0.524823
\(900\) 5.44574e9 0.249005
\(901\) −7.84539e9 −0.357337
\(902\) −5.12803e9 −0.232663
\(903\) 2.34595e8 0.0106026
\(904\) 1.27807e10 0.575393
\(905\) −1.19261e9 −0.0534844
\(906\) 6.76915e9 0.302403
\(907\) −2.36818e10 −1.05388 −0.526938 0.849904i \(-0.676660\pi\)
−0.526938 + 0.849904i \(0.676660\pi\)
\(908\) −4.27556e9 −0.189536
\(909\) 2.30628e9 0.101845
\(910\) 4.87074e6 0.000214265 0
\(911\) 1.78985e10 0.784338 0.392169 0.919893i \(-0.371725\pi\)
0.392169 + 0.919893i \(0.371725\pi\)
\(912\) 7.42566e8 0.0324155
\(913\) −9.97853e8 −0.0433929
\(914\) 1.81295e9 0.0785371
\(915\) 3.56249e8 0.0153738
\(916\) 9.20650e9 0.395786
\(917\) 2.04035e10 0.873799
\(918\) 1.87055e9 0.0798032
\(919\) −5.16659e9 −0.219584 −0.109792 0.993955i \(-0.535018\pi\)
−0.109792 + 0.993955i \(0.535018\pi\)
\(920\) 1.42254e9 0.0602294
\(921\) 1.26295e10 0.532694
\(922\) 5.41298e9 0.227446
\(923\) 8.28317e7 0.00346729
\(924\) −6.78899e9 −0.283109
\(925\) −2.48607e10 −1.03280
\(926\) −2.01225e10 −0.832804
\(927\) −6.47481e9 −0.266961
\(928\) −1.12256e10 −0.461095
\(929\) 2.17904e10 0.891681 0.445840 0.895112i \(-0.352905\pi\)
0.445840 + 0.895112i \(0.352905\pi\)
\(930\) 4.47316e8 0.0182358
\(931\) −3.10475e9 −0.126096
\(932\) 1.43769e10 0.581713
\(933\) −9.36371e8 −0.0377452
\(934\) −2.31656e10 −0.930315
\(935\) 1.32231e9 0.0529047
\(936\) 1.06037e8 0.00422662
\(937\) −2.40380e10 −0.954575 −0.477287 0.878747i \(-0.658380\pi\)
−0.477287 + 0.878747i \(0.658380\pi\)
\(938\) 8.52538e8 0.0337290
\(939\) 8.82539e9 0.347860
\(940\) 1.01057e9 0.0396843
\(941\) 4.36782e10 1.70884 0.854419 0.519585i \(-0.173913\pi\)
0.854419 + 0.519585i \(0.173913\pi\)
\(942\) 9.05077e9 0.352783
\(943\) −1.28721e10 −0.499872
\(944\) 1.04462e9 0.0404163
\(945\) −1.47570e8 −0.00568837
\(946\) −5.17705e8 −0.0198821
\(947\) 1.94958e10 0.745961 0.372980 0.927839i \(-0.378336\pi\)
0.372980 + 0.927839i \(0.378336\pi\)
\(948\) −9.80564e9 −0.373806
\(949\) 7.49459e8 0.0284653
\(950\) 2.38667e9 0.0903151
\(951\) 1.08044e10 0.407350
\(952\) −1.06250e10 −0.399115
\(953\) 1.31470e10 0.492040 0.246020 0.969265i \(-0.420877\pi\)
0.246020 + 0.969265i \(0.420877\pi\)
\(954\) −1.93215e9 −0.0720480
\(955\) −2.07175e9 −0.0769707
\(956\) 4.79079e9 0.177339
\(957\) 8.32374e9 0.306992
\(958\) −1.25402e10 −0.460813
\(959\) 2.16875e10 0.794045
\(960\) −1.75265e8 −0.00639359
\(961\) 1.04117e10 0.378435
\(962\) −2.07331e8 −0.00750848
\(963\) 1.22531e10 0.442135
\(964\) −3.77877e8 −0.0135857
\(965\) −1.30835e7 −0.000468682 0
\(966\) 5.70542e9 0.203642
\(967\) 4.78218e10 1.70072 0.850361 0.526200i \(-0.176384\pi\)
0.850361 + 0.526200i \(0.176384\pi\)
\(968\) 1.02579e10 0.363493
\(969\) −2.44863e9 −0.0864548
\(970\) 1.25371e9 0.0441059
\(971\) 2.03563e10 0.713560 0.356780 0.934188i \(-0.383875\pi\)
0.356780 + 0.934188i \(0.383875\pi\)
\(972\) −1.37598e9 −0.0480597
\(973\) −1.49824e10 −0.521418
\(974\) 2.65374e10 0.920242
\(975\) −2.41156e8 −0.00833262
\(976\) −4.46982e9 −0.153892
\(977\) −4.11493e10 −1.41166 −0.705832 0.708379i \(-0.749427\pi\)
−0.705832 + 0.708379i \(0.749427\pi\)
\(978\) 5.29152e9 0.180882
\(979\) 1.42082e10 0.483949
\(980\) −8.26713e8 −0.0280584
\(981\) 4.52183e9 0.152923
\(982\) 1.01935e10 0.343506
\(983\) −2.78989e10 −0.936805 −0.468402 0.883515i \(-0.655170\pi\)
−0.468402 + 0.883515i \(0.655170\pi\)
\(984\) 5.90358e9 0.197530
\(985\) −2.57992e9 −0.0860160
\(986\) 5.57944e9 0.185362
\(987\) 9.46320e9 0.313277
\(988\) −5.94514e7 −0.00196116
\(989\) −1.29952e9 −0.0427164
\(990\) 3.25658e8 0.0106669
\(991\) −5.57475e10 −1.81957 −0.909783 0.415084i \(-0.863752\pi\)
−0.909783 + 0.415084i \(0.863752\pi\)
\(992\) −3.72353e10 −1.21106
\(993\) −5.62846e9 −0.182418
\(994\) −2.04405e9 −0.0660144
\(995\) 1.42953e9 0.0460058
\(996\) 4.92019e8 0.0157788
\(997\) 1.05869e10 0.338326 0.169163 0.985588i \(-0.445894\pi\)
0.169163 + 0.985588i \(0.445894\pi\)
\(998\) 4.82724e9 0.153724
\(999\) 6.28158e9 0.199338
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.1.7 18
3.2 odd 2 531.8.a.e.1.12 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.7 18 1.1 even 1 trivial
531.8.a.e.1.12 18 3.2 odd 2