Properties

Label 177.8.a.b.1.15
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 1.15
Root \(17.5255\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+15.5255 q^{2} +27.0000 q^{3} +113.041 q^{4} -141.845 q^{5} +419.189 q^{6} +105.285 q^{7} -232.240 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+15.5255 q^{2} +27.0000 q^{3} +113.041 q^{4} -141.845 q^{5} +419.189 q^{6} +105.285 q^{7} -232.240 q^{8} +729.000 q^{9} -2202.21 q^{10} -555.209 q^{11} +3052.12 q^{12} -11289.1 q^{13} +1634.61 q^{14} -3829.81 q^{15} -18074.9 q^{16} -3139.99 q^{17} +11318.1 q^{18} -21947.5 q^{19} -16034.3 q^{20} +2842.71 q^{21} -8619.91 q^{22} +28575.9 q^{23} -6270.49 q^{24} -58005.1 q^{25} -175269. q^{26} +19683.0 q^{27} +11901.6 q^{28} +160286. q^{29} -59459.7 q^{30} -163594. q^{31} -250896. q^{32} -14990.7 q^{33} -48750.0 q^{34} -14934.2 q^{35} +82407.2 q^{36} -131107. q^{37} -340746. q^{38} -304806. q^{39} +32942.1 q^{40} -277081. q^{41} +44134.5 q^{42} +621893. q^{43} -62761.6 q^{44} -103405. q^{45} +443655. q^{46} -1.39412e6 q^{47} -488023. q^{48} -812458. q^{49} -900558. q^{50} -84779.8 q^{51} -1.27614e6 q^{52} +1.49504e6 q^{53} +305589. q^{54} +78753.5 q^{55} -24451.5 q^{56} -592583. q^{57} +2.48851e6 q^{58} -205379. q^{59} -432927. q^{60} -2.41889e6 q^{61} -2.53988e6 q^{62} +76753.1 q^{63} -1.58169e6 q^{64} +1.60130e6 q^{65} -232737. q^{66} +2.34039e6 q^{67} -354949. q^{68} +771549. q^{69} -231861. q^{70} +52758.6 q^{71} -169303. q^{72} +1.12536e6 q^{73} -2.03550e6 q^{74} -1.56614e6 q^{75} -2.48098e6 q^{76} -58455.5 q^{77} -4.73227e6 q^{78} +3.33432e6 q^{79} +2.56384e6 q^{80} +531441. q^{81} -4.30183e6 q^{82} -5.69160e6 q^{83} +321343. q^{84} +445391. q^{85} +9.65520e6 q^{86} +4.32771e6 q^{87} +128942. q^{88} -5.02101e6 q^{89} -1.60541e6 q^{90} -1.18858e6 q^{91} +3.23026e6 q^{92} -4.41704e6 q^{93} -2.16444e7 q^{94} +3.11314e6 q^{95} -6.77419e6 q^{96} +9.04920e6 q^{97} -1.26138e7 q^{98} -404748. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 15.5255 1.37227 0.686137 0.727472i \(-0.259305\pi\)
0.686137 + 0.727472i \(0.259305\pi\)
\(3\) 27.0000 0.577350
\(4\) 113.041 0.883136
\(5\) −141.845 −0.507479 −0.253740 0.967273i \(-0.581661\pi\)
−0.253740 + 0.967273i \(0.581661\pi\)
\(6\) 419.189 0.792283
\(7\) 105.285 0.116018 0.0580090 0.998316i \(-0.481525\pi\)
0.0580090 + 0.998316i \(0.481525\pi\)
\(8\) −232.240 −0.160370
\(9\) 729.000 0.333333
\(10\) −2202.21 −0.696401
\(11\) −555.209 −0.125772 −0.0628858 0.998021i \(-0.520030\pi\)
−0.0628858 + 0.998021i \(0.520030\pi\)
\(12\) 3052.12 0.509879
\(13\) −11289.1 −1.42514 −0.712572 0.701599i \(-0.752470\pi\)
−0.712572 + 0.701599i \(0.752470\pi\)
\(14\) 1634.61 0.159208
\(15\) −3829.81 −0.292993
\(16\) −18074.9 −1.10321
\(17\) −3139.99 −0.155009 −0.0775046 0.996992i \(-0.524695\pi\)
−0.0775046 + 0.996992i \(0.524695\pi\)
\(18\) 11318.1 0.457425
\(19\) −21947.5 −0.734088 −0.367044 0.930204i \(-0.619630\pi\)
−0.367044 + 0.930204i \(0.619630\pi\)
\(20\) −16034.3 −0.448173
\(21\) 2842.71 0.0669830
\(22\) −8619.91 −0.172593
\(23\) 28575.9 0.489725 0.244863 0.969558i \(-0.421257\pi\)
0.244863 + 0.969558i \(0.421257\pi\)
\(24\) −6270.49 −0.0925895
\(25\) −58005.1 −0.742465
\(26\) −175269. −1.95569
\(27\) 19683.0 0.192450
\(28\) 11901.6 0.102460
\(29\) 160286. 1.22040 0.610199 0.792248i \(-0.291090\pi\)
0.610199 + 0.792248i \(0.291090\pi\)
\(30\) −59459.7 −0.402067
\(31\) −163594. −0.986284 −0.493142 0.869949i \(-0.664152\pi\)
−0.493142 + 0.869949i \(0.664152\pi\)
\(32\) −250896. −1.35353
\(33\) −14990.7 −0.0726142
\(34\) −48750.0 −0.212715
\(35\) −14934.2 −0.0588767
\(36\) 82407.2 0.294379
\(37\) −131107. −0.425519 −0.212759 0.977105i \(-0.568245\pi\)
−0.212759 + 0.977105i \(0.568245\pi\)
\(38\) −340746. −1.00737
\(39\) −304806. −0.822807
\(40\) 32942.1 0.0813843
\(41\) −277081. −0.627862 −0.313931 0.949446i \(-0.601646\pi\)
−0.313931 + 0.949446i \(0.601646\pi\)
\(42\) 44134.5 0.0919190
\(43\) 621893. 1.19282 0.596411 0.802679i \(-0.296593\pi\)
0.596411 + 0.802679i \(0.296593\pi\)
\(44\) −62761.6 −0.111073
\(45\) −103405. −0.169160
\(46\) 443655. 0.672037
\(47\) −1.39412e6 −1.95865 −0.979325 0.202292i \(-0.935161\pi\)
−0.979325 + 0.202292i \(0.935161\pi\)
\(48\) −488023. −0.636937
\(49\) −812458. −0.986540
\(50\) −900558. −1.01887
\(51\) −84779.8 −0.0894946
\(52\) −1.27614e6 −1.25860
\(53\) 1.49504e6 1.37939 0.689696 0.724099i \(-0.257744\pi\)
0.689696 + 0.724099i \(0.257744\pi\)
\(54\) 305589. 0.264094
\(55\) 78753.5 0.0638265
\(56\) −24451.5 −0.0186058
\(57\) −592583. −0.423826
\(58\) 2.48851e6 1.67472
\(59\) −205379. −0.130189
\(60\) −432927. −0.258753
\(61\) −2.41889e6 −1.36446 −0.682230 0.731137i \(-0.738990\pi\)
−0.682230 + 0.731137i \(0.738990\pi\)
\(62\) −2.53988e6 −1.35345
\(63\) 76753.1 0.0386726
\(64\) −1.58169e6 −0.754210
\(65\) 1.60130e6 0.723231
\(66\) −232737. −0.0996466
\(67\) 2.34039e6 0.950664 0.475332 0.879807i \(-0.342328\pi\)
0.475332 + 0.879807i \(0.342328\pi\)
\(68\) −354949. −0.136894
\(69\) 771549. 0.282743
\(70\) −231861. −0.0807950
\(71\) 52758.6 0.0174940 0.00874700 0.999962i \(-0.497216\pi\)
0.00874700 + 0.999962i \(0.497216\pi\)
\(72\) −169303. −0.0534566
\(73\) 1.12536e6 0.338581 0.169290 0.985566i \(-0.445852\pi\)
0.169290 + 0.985566i \(0.445852\pi\)
\(74\) −2.03550e6 −0.583928
\(75\) −1.56614e6 −0.428662
\(76\) −2.48098e6 −0.648299
\(77\) −58455.5 −0.0145918
\(78\) −4.73227e6 −1.12912
\(79\) 3.33432e6 0.760873 0.380436 0.924807i \(-0.375774\pi\)
0.380436 + 0.924807i \(0.375774\pi\)
\(80\) 2.56384e6 0.559855
\(81\) 531441. 0.111111
\(82\) −4.30183e6 −0.861598
\(83\) −5.69160e6 −1.09260 −0.546300 0.837590i \(-0.683964\pi\)
−0.546300 + 0.837590i \(0.683964\pi\)
\(84\) 321343. 0.0591551
\(85\) 445391. 0.0786639
\(86\) 9.65520e6 1.63688
\(87\) 4.32771e6 0.704597
\(88\) 128942. 0.0201700
\(89\) −5.02101e6 −0.754963 −0.377482 0.926017i \(-0.623210\pi\)
−0.377482 + 0.926017i \(0.623210\pi\)
\(90\) −1.60541e6 −0.232134
\(91\) −1.18858e6 −0.165342
\(92\) 3.23026e6 0.432494
\(93\) −4.41704e6 −0.569431
\(94\) −2.16444e7 −2.68781
\(95\) 3.11314e6 0.372534
\(96\) −6.77419e6 −0.781462
\(97\) 9.04920e6 1.00672 0.503361 0.864076i \(-0.332097\pi\)
0.503361 + 0.864076i \(0.332097\pi\)
\(98\) −1.26138e7 −1.35380
\(99\) −404748. −0.0419238
\(100\) −6.55697e6 −0.655697
\(101\) 1.74915e6 0.168928 0.0844639 0.996427i \(-0.473082\pi\)
0.0844639 + 0.996427i \(0.473082\pi\)
\(102\) −1.31625e6 −0.122811
\(103\) 1.68825e7 1.52232 0.761160 0.648564i \(-0.224630\pi\)
0.761160 + 0.648564i \(0.224630\pi\)
\(104\) 2.62179e6 0.228550
\(105\) −403223. −0.0339925
\(106\) 2.32113e7 1.89290
\(107\) 7.69410e6 0.607176 0.303588 0.952803i \(-0.401815\pi\)
0.303588 + 0.952803i \(0.401815\pi\)
\(108\) 2.22499e6 0.169960
\(109\) −2.31641e7 −1.71326 −0.856629 0.515932i \(-0.827446\pi\)
−0.856629 + 0.515932i \(0.827446\pi\)
\(110\) 1.22269e6 0.0875874
\(111\) −3.53988e6 −0.245673
\(112\) −1.90303e6 −0.127992
\(113\) 2.01725e7 1.31518 0.657590 0.753376i \(-0.271576\pi\)
0.657590 + 0.753376i \(0.271576\pi\)
\(114\) −9.20015e6 −0.581605
\(115\) −4.05334e6 −0.248525
\(116\) 1.81189e7 1.07778
\(117\) −8.22977e6 −0.475048
\(118\) −3.18861e6 −0.178655
\(119\) −330595. −0.0179838
\(120\) 889436. 0.0469873
\(121\) −1.91789e7 −0.984182
\(122\) −3.75544e7 −1.87241
\(123\) −7.48120e6 −0.362496
\(124\) −1.84929e7 −0.871023
\(125\) 1.93093e7 0.884265
\(126\) 1.19163e6 0.0530695
\(127\) 1.39057e6 0.0602395 0.0301197 0.999546i \(-0.490411\pi\)
0.0301197 + 0.999546i \(0.490411\pi\)
\(128\) 7.55808e6 0.318549
\(129\) 1.67911e7 0.688677
\(130\) 2.48611e7 0.992471
\(131\) −4.16332e6 −0.161804 −0.0809022 0.996722i \(-0.525780\pi\)
−0.0809022 + 0.996722i \(0.525780\pi\)
\(132\) −1.69456e6 −0.0641282
\(133\) −2.31075e6 −0.0851673
\(134\) 3.63358e7 1.30457
\(135\) −2.79193e6 −0.0976644
\(136\) 729232. 0.0248588
\(137\) 3.71579e7 1.23461 0.617304 0.786724i \(-0.288225\pi\)
0.617304 + 0.786724i \(0.288225\pi\)
\(138\) 1.19787e7 0.388001
\(139\) −738784. −0.0233327 −0.0116664 0.999932i \(-0.503714\pi\)
−0.0116664 + 0.999932i \(0.503714\pi\)
\(140\) −1.68818e6 −0.0519961
\(141\) −3.76412e7 −1.13083
\(142\) 819104. 0.0240066
\(143\) 6.26783e6 0.179243
\(144\) −1.31766e7 −0.367736
\(145\) −2.27357e7 −0.619327
\(146\) 1.74718e7 0.464626
\(147\) −2.19364e7 −0.569579
\(148\) −1.48205e7 −0.375791
\(149\) −7.16242e7 −1.77381 −0.886906 0.461949i \(-0.847150\pi\)
−0.886906 + 0.461949i \(0.847150\pi\)
\(150\) −2.43151e7 −0.588242
\(151\) −3.49956e7 −0.827168 −0.413584 0.910466i \(-0.635723\pi\)
−0.413584 + 0.910466i \(0.635723\pi\)
\(152\) 5.09710e6 0.117725
\(153\) −2.28905e6 −0.0516697
\(154\) −907551. −0.0200239
\(155\) 2.32050e7 0.500519
\(156\) −3.44557e7 −0.726650
\(157\) −3.84270e7 −0.792478 −0.396239 0.918147i \(-0.629685\pi\)
−0.396239 + 0.918147i \(0.629685\pi\)
\(158\) 5.17669e7 1.04413
\(159\) 4.03661e7 0.796393
\(160\) 3.55883e7 0.686890
\(161\) 3.00863e6 0.0568169
\(162\) 8.25089e6 0.152475
\(163\) −4.69770e7 −0.849627 −0.424813 0.905281i \(-0.639660\pi\)
−0.424813 + 0.905281i \(0.639660\pi\)
\(164\) −3.13217e7 −0.554487
\(165\) 2.12635e6 0.0368502
\(166\) −8.83650e7 −1.49935
\(167\) 5.23525e7 0.869820 0.434910 0.900474i \(-0.356780\pi\)
0.434910 + 0.900474i \(0.356780\pi\)
\(168\) −660191. −0.0107420
\(169\) 6.46959e7 1.03103
\(170\) 6.91493e6 0.107948
\(171\) −1.59997e7 −0.244696
\(172\) 7.02996e7 1.05342
\(173\) −1.05438e8 −1.54823 −0.774115 0.633046i \(-0.781805\pi\)
−0.774115 + 0.633046i \(0.781805\pi\)
\(174\) 6.71899e7 0.966900
\(175\) −6.10709e6 −0.0861392
\(176\) 1.00354e7 0.138752
\(177\) −5.54523e6 −0.0751646
\(178\) −7.79537e7 −1.03602
\(179\) −7.74219e7 −1.00897 −0.504485 0.863421i \(-0.668318\pi\)
−0.504485 + 0.863421i \(0.668318\pi\)
\(180\) −1.16890e7 −0.149391
\(181\) 1.44384e8 1.80986 0.904930 0.425561i \(-0.139923\pi\)
0.904930 + 0.425561i \(0.139923\pi\)
\(182\) −1.84533e7 −0.226895
\(183\) −6.53099e7 −0.787772
\(184\) −6.63647e6 −0.0785371
\(185\) 1.85968e7 0.215942
\(186\) −6.85768e7 −0.781416
\(187\) 1.74335e6 0.0194957
\(188\) −1.57593e8 −1.72975
\(189\) 2.07233e6 0.0223277
\(190\) 4.83331e7 0.511219
\(191\) 1.83193e8 1.90236 0.951178 0.308642i \(-0.0998746\pi\)
0.951178 + 0.308642i \(0.0998746\pi\)
\(192\) −4.27057e7 −0.435444
\(193\) 9.07393e7 0.908542 0.454271 0.890864i \(-0.349900\pi\)
0.454271 + 0.890864i \(0.349900\pi\)
\(194\) 1.40493e8 1.38150
\(195\) 4.32352e7 0.417558
\(196\) −9.18414e7 −0.871249
\(197\) 9.27275e7 0.864125 0.432063 0.901844i \(-0.357786\pi\)
0.432063 + 0.901844i \(0.357786\pi\)
\(198\) −6.28391e6 −0.0575310
\(199\) 7.94761e7 0.714909 0.357455 0.933931i \(-0.383645\pi\)
0.357455 + 0.933931i \(0.383645\pi\)
\(200\) 1.34711e7 0.119069
\(201\) 6.31906e7 0.548866
\(202\) 2.71564e7 0.231815
\(203\) 1.68757e7 0.141588
\(204\) −9.58362e6 −0.0790358
\(205\) 3.93026e7 0.318627
\(206\) 2.62109e8 2.08904
\(207\) 2.08318e7 0.163242
\(208\) 2.04050e8 1.57223
\(209\) 1.21855e7 0.0923273
\(210\) −6.26024e6 −0.0466470
\(211\) 1.26710e8 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(212\) 1.69002e8 1.21819
\(213\) 1.42448e6 0.0101002
\(214\) 1.19455e8 0.833212
\(215\) −8.82123e7 −0.605333
\(216\) −4.57118e6 −0.0308632
\(217\) −1.72241e7 −0.114427
\(218\) −3.59635e8 −2.35106
\(219\) 3.03848e7 0.195480
\(220\) 8.90241e6 0.0563674
\(221\) 3.54478e7 0.220910
\(222\) −5.49584e7 −0.337131
\(223\) 1.08372e8 0.654410 0.327205 0.944953i \(-0.393893\pi\)
0.327205 + 0.944953i \(0.393893\pi\)
\(224\) −2.64157e7 −0.157034
\(225\) −4.22857e7 −0.247488
\(226\) 3.13188e8 1.80479
\(227\) 1.82593e8 1.03608 0.518040 0.855356i \(-0.326662\pi\)
0.518040 + 0.855356i \(0.326662\pi\)
\(228\) −6.69864e7 −0.374296
\(229\) 1.61240e8 0.887256 0.443628 0.896211i \(-0.353691\pi\)
0.443628 + 0.896211i \(0.353691\pi\)
\(230\) −6.29302e7 −0.341045
\(231\) −1.57830e6 −0.00842455
\(232\) −3.72247e7 −0.195715
\(233\) −1.36205e8 −0.705421 −0.352711 0.935732i \(-0.614740\pi\)
−0.352711 + 0.935732i \(0.614740\pi\)
\(234\) −1.27771e8 −0.651896
\(235\) 1.97748e8 0.993975
\(236\) −2.32163e7 −0.114974
\(237\) 9.00265e7 0.439290
\(238\) −5.13266e6 −0.0246788
\(239\) −1.32804e8 −0.629241 −0.314620 0.949218i \(-0.601877\pi\)
−0.314620 + 0.949218i \(0.601877\pi\)
\(240\) 6.92236e7 0.323232
\(241\) −1.82912e8 −0.841749 −0.420874 0.907119i \(-0.638277\pi\)
−0.420874 + 0.907119i \(0.638277\pi\)
\(242\) −2.97762e8 −1.35057
\(243\) 1.43489e7 0.0641500
\(244\) −2.73434e8 −1.20500
\(245\) 1.15243e8 0.500649
\(246\) −1.16149e8 −0.497444
\(247\) 2.47768e8 1.04618
\(248\) 3.79931e7 0.158170
\(249\) −1.53673e8 −0.630813
\(250\) 2.99787e8 1.21345
\(251\) −3.69029e8 −1.47300 −0.736500 0.676438i \(-0.763523\pi\)
−0.736500 + 0.676438i \(0.763523\pi\)
\(252\) 8.67627e6 0.0341532
\(253\) −1.58656e7 −0.0615935
\(254\) 2.15894e7 0.0826651
\(255\) 1.20256e7 0.0454166
\(256\) 3.19800e8 1.19135
\(257\) −2.11987e8 −0.779010 −0.389505 0.921024i \(-0.627354\pi\)
−0.389505 + 0.921024i \(0.627354\pi\)
\(258\) 2.60690e8 0.945053
\(259\) −1.38036e7 −0.0493678
\(260\) 1.81014e8 0.638711
\(261\) 1.16848e8 0.406799
\(262\) −6.46377e7 −0.222040
\(263\) −1.26702e8 −0.429476 −0.214738 0.976672i \(-0.568890\pi\)
−0.214738 + 0.976672i \(0.568890\pi\)
\(264\) 3.48143e6 0.0116451
\(265\) −2.12064e8 −0.700013
\(266\) −3.58756e7 −0.116873
\(267\) −1.35567e8 −0.435878
\(268\) 2.64561e8 0.839565
\(269\) −2.67421e8 −0.837650 −0.418825 0.908067i \(-0.637558\pi\)
−0.418825 + 0.908067i \(0.637558\pi\)
\(270\) −4.33461e7 −0.134022
\(271\) −2.12216e7 −0.0647718 −0.0323859 0.999475i \(-0.510311\pi\)
−0.0323859 + 0.999475i \(0.510311\pi\)
\(272\) 5.67552e7 0.171007
\(273\) −3.20917e7 −0.0954604
\(274\) 5.76895e8 1.69422
\(275\) 3.22050e7 0.0933809
\(276\) 8.72170e7 0.249700
\(277\) 2.91983e8 0.825425 0.412713 0.910861i \(-0.364581\pi\)
0.412713 + 0.910861i \(0.364581\pi\)
\(278\) −1.14700e7 −0.0320189
\(279\) −1.19260e8 −0.328761
\(280\) 3.46832e6 0.00944204
\(281\) 2.59836e8 0.698597 0.349298 0.937012i \(-0.386420\pi\)
0.349298 + 0.937012i \(0.386420\pi\)
\(282\) −5.84398e8 −1.55180
\(283\) 3.61529e8 0.948180 0.474090 0.880476i \(-0.342777\pi\)
0.474090 + 0.880476i \(0.342777\pi\)
\(284\) 5.96391e6 0.0154496
\(285\) 8.40548e7 0.215083
\(286\) 9.73112e7 0.245970
\(287\) −2.91726e7 −0.0728432
\(288\) −1.82903e8 −0.451177
\(289\) −4.00479e8 −0.975972
\(290\) −3.52983e8 −0.849886
\(291\) 2.44328e8 0.581231
\(292\) 1.27213e8 0.299013
\(293\) −6.52155e8 −1.51466 −0.757328 0.653034i \(-0.773496\pi\)
−0.757328 + 0.653034i \(0.773496\pi\)
\(294\) −3.40573e8 −0.781618
\(295\) 2.91319e7 0.0660682
\(296\) 3.04482e7 0.0682404
\(297\) −1.09282e7 −0.0242047
\(298\) −1.11200e9 −2.43416
\(299\) −3.22597e8 −0.697929
\(300\) −1.77038e8 −0.378567
\(301\) 6.54763e7 0.138389
\(302\) −5.43324e8 −1.13510
\(303\) 4.72269e7 0.0975305
\(304\) 3.96700e8 0.809851
\(305\) 3.43106e8 0.692436
\(306\) −3.55387e7 −0.0709050
\(307\) −5.34821e8 −1.05493 −0.527466 0.849576i \(-0.676858\pi\)
−0.527466 + 0.849576i \(0.676858\pi\)
\(308\) −6.60789e6 −0.0128865
\(309\) 4.55827e8 0.878912
\(310\) 3.60269e8 0.686849
\(311\) −8.03720e8 −1.51511 −0.757554 0.652773i \(-0.773606\pi\)
−0.757554 + 0.652773i \(0.773606\pi\)
\(312\) 7.07883e7 0.131953
\(313\) −3.67380e8 −0.677190 −0.338595 0.940932i \(-0.609952\pi\)
−0.338595 + 0.940932i \(0.609952\pi\)
\(314\) −5.96598e8 −1.08750
\(315\) −1.08870e7 −0.0196256
\(316\) 3.76916e8 0.671954
\(317\) −6.78407e8 −1.19614 −0.598071 0.801443i \(-0.704066\pi\)
−0.598071 + 0.801443i \(0.704066\pi\)
\(318\) 6.26705e8 1.09287
\(319\) −8.89920e7 −0.153491
\(320\) 2.24355e8 0.382746
\(321\) 2.07741e8 0.350553
\(322\) 4.67105e7 0.0779684
\(323\) 6.89150e7 0.113790
\(324\) 6.00748e7 0.0981262
\(325\) 6.54827e8 1.05812
\(326\) −7.29341e8 −1.16592
\(327\) −6.25431e8 −0.989151
\(328\) 6.43494e7 0.100690
\(329\) −1.46780e8 −0.227239
\(330\) 3.30126e7 0.0505686
\(331\) −1.80151e8 −0.273048 −0.136524 0.990637i \(-0.543593\pi\)
−0.136524 + 0.990637i \(0.543593\pi\)
\(332\) −6.43386e8 −0.964914
\(333\) −9.55768e7 −0.141840
\(334\) 8.12799e8 1.19363
\(335\) −3.31973e8 −0.482442
\(336\) −5.13818e7 −0.0738961
\(337\) −6.10617e8 −0.869089 −0.434544 0.900650i \(-0.643091\pi\)
−0.434544 + 0.900650i \(0.643091\pi\)
\(338\) 1.00444e9 1.41486
\(339\) 5.44658e8 0.759320
\(340\) 5.03477e7 0.0694709
\(341\) 9.08290e7 0.124046
\(342\) −2.48404e8 −0.335790
\(343\) −1.72247e8 −0.230474
\(344\) −1.44429e8 −0.191293
\(345\) −1.09440e8 −0.143486
\(346\) −1.63698e9 −2.12459
\(347\) −1.97837e8 −0.254188 −0.127094 0.991891i \(-0.540565\pi\)
−0.127094 + 0.991891i \(0.540565\pi\)
\(348\) 4.89210e8 0.622255
\(349\) −9.01885e8 −1.13570 −0.567848 0.823133i \(-0.692224\pi\)
−0.567848 + 0.823133i \(0.692224\pi\)
\(350\) −9.48156e7 −0.118207
\(351\) −2.22204e8 −0.274269
\(352\) 1.39300e8 0.170236
\(353\) −8.58268e8 −1.03851 −0.519256 0.854619i \(-0.673791\pi\)
−0.519256 + 0.854619i \(0.673791\pi\)
\(354\) −8.60926e7 −0.103146
\(355\) −7.48353e6 −0.00887784
\(356\) −5.67582e8 −0.666735
\(357\) −8.92607e6 −0.0103830
\(358\) −1.20201e9 −1.38458
\(359\) −4.75915e8 −0.542874 −0.271437 0.962456i \(-0.587499\pi\)
−0.271437 + 0.962456i \(0.587499\pi\)
\(360\) 2.40148e7 0.0271281
\(361\) −4.12178e8 −0.461115
\(362\) 2.24164e9 2.48362
\(363\) −5.17831e8 −0.568217
\(364\) −1.34359e8 −0.146020
\(365\) −1.59627e8 −0.171823
\(366\) −1.01397e9 −1.08104
\(367\) 7.17854e8 0.758062 0.379031 0.925384i \(-0.376257\pi\)
0.379031 + 0.925384i \(0.376257\pi\)
\(368\) −5.16508e8 −0.540268
\(369\) −2.01992e8 −0.209287
\(370\) 2.88725e8 0.296332
\(371\) 1.57406e8 0.160034
\(372\) −4.99309e8 −0.502885
\(373\) −5.85967e8 −0.584645 −0.292322 0.956320i \(-0.594428\pi\)
−0.292322 + 0.956320i \(0.594428\pi\)
\(374\) 2.70664e7 0.0267535
\(375\) 5.21352e8 0.510530
\(376\) 3.23770e8 0.314108
\(377\) −1.80948e9 −1.73924
\(378\) 3.21740e7 0.0306397
\(379\) 1.74459e8 0.164610 0.0823050 0.996607i \(-0.473772\pi\)
0.0823050 + 0.996607i \(0.473772\pi\)
\(380\) 3.51914e8 0.328998
\(381\) 3.75455e7 0.0347793
\(382\) 2.84416e9 2.61055
\(383\) −6.24930e8 −0.568376 −0.284188 0.958769i \(-0.591724\pi\)
−0.284188 + 0.958769i \(0.591724\pi\)
\(384\) 2.04068e8 0.183915
\(385\) 8.29160e6 0.00740501
\(386\) 1.40877e9 1.24677
\(387\) 4.53360e8 0.397608
\(388\) 1.02293e9 0.889071
\(389\) −2.05004e8 −0.176579 −0.0882896 0.996095i \(-0.528140\pi\)
−0.0882896 + 0.996095i \(0.528140\pi\)
\(390\) 6.71248e8 0.573003
\(391\) −8.97281e7 −0.0759119
\(392\) 1.88685e8 0.158211
\(393\) −1.12410e8 −0.0934178
\(394\) 1.43964e9 1.18582
\(395\) −4.72955e8 −0.386127
\(396\) −4.57532e7 −0.0370245
\(397\) −1.96710e9 −1.57783 −0.788915 0.614502i \(-0.789357\pi\)
−0.788915 + 0.614502i \(0.789357\pi\)
\(398\) 1.23391e9 0.981051
\(399\) −6.23904e7 −0.0491714
\(400\) 1.04844e9 0.819092
\(401\) −2.09175e9 −1.61996 −0.809979 0.586458i \(-0.800522\pi\)
−0.809979 + 0.586458i \(0.800522\pi\)
\(402\) 9.81066e8 0.753195
\(403\) 1.84684e9 1.40560
\(404\) 1.97726e8 0.149186
\(405\) −7.53821e7 −0.0563866
\(406\) 2.62004e8 0.194298
\(407\) 7.27917e7 0.0535182
\(408\) 1.96893e7 0.0143522
\(409\) 3.66808e8 0.265099 0.132549 0.991176i \(-0.457684\pi\)
0.132549 + 0.991176i \(0.457684\pi\)
\(410\) 6.10192e8 0.437243
\(411\) 1.00326e9 0.712802
\(412\) 1.90842e9 1.34441
\(413\) −2.16234e7 −0.0151042
\(414\) 3.23425e8 0.224012
\(415\) 8.07324e8 0.554472
\(416\) 2.83240e9 1.92898
\(417\) −1.99472e7 −0.0134712
\(418\) 1.89186e8 0.126698
\(419\) 1.27304e9 0.845463 0.422731 0.906255i \(-0.361071\pi\)
0.422731 + 0.906255i \(0.361071\pi\)
\(420\) −4.55809e7 −0.0300200
\(421\) −6.98288e8 −0.456087 −0.228043 0.973651i \(-0.573233\pi\)
−0.228043 + 0.973651i \(0.573233\pi\)
\(422\) 1.96724e9 1.27428
\(423\) −1.01631e9 −0.652884
\(424\) −3.47209e8 −0.221213
\(425\) 1.82135e8 0.115089
\(426\) 2.21158e7 0.0138602
\(427\) −2.54674e8 −0.158302
\(428\) 8.69752e8 0.536219
\(429\) 1.69231e8 0.103486
\(430\) −1.36954e9 −0.830683
\(431\) 2.63872e8 0.158753 0.0793765 0.996845i \(-0.474707\pi\)
0.0793765 + 0.996845i \(0.474707\pi\)
\(432\) −3.55769e8 −0.212312
\(433\) 2.13875e9 1.26606 0.633028 0.774128i \(-0.281812\pi\)
0.633028 + 0.774128i \(0.281812\pi\)
\(434\) −2.67413e8 −0.157025
\(435\) −6.13863e8 −0.357568
\(436\) −2.61850e9 −1.51304
\(437\) −6.27170e8 −0.359501
\(438\) 4.71739e8 0.268252
\(439\) 2.80384e9 1.58171 0.790856 0.612003i \(-0.209636\pi\)
0.790856 + 0.612003i \(0.209636\pi\)
\(440\) −1.82897e7 −0.0102358
\(441\) −5.92282e8 −0.328847
\(442\) 5.50345e8 0.303149
\(443\) 2.51161e9 1.37259 0.686293 0.727326i \(-0.259237\pi\)
0.686293 + 0.727326i \(0.259237\pi\)
\(444\) −4.00153e8 −0.216963
\(445\) 7.12204e8 0.383128
\(446\) 1.68253e9 0.898030
\(447\) −1.93385e9 −1.02411
\(448\) −1.66529e8 −0.0875019
\(449\) 2.93296e9 1.52913 0.764564 0.644548i \(-0.222955\pi\)
0.764564 + 0.644548i \(0.222955\pi\)
\(450\) −6.56507e8 −0.339622
\(451\) 1.53838e8 0.0789671
\(452\) 2.28033e9 1.16148
\(453\) −9.44880e8 −0.477566
\(454\) 2.83485e9 1.42179
\(455\) 1.68594e8 0.0839078
\(456\) 1.37622e8 0.0679688
\(457\) −3.30374e9 −1.61920 −0.809598 0.586984i \(-0.800315\pi\)
−0.809598 + 0.586984i \(0.800315\pi\)
\(458\) 2.50333e9 1.21756
\(459\) −6.18045e7 −0.0298315
\(460\) −4.58195e8 −0.219482
\(461\) 6.28637e7 0.0298845 0.0149423 0.999888i \(-0.495244\pi\)
0.0149423 + 0.999888i \(0.495244\pi\)
\(462\) −2.45039e7 −0.0115608
\(463\) −2.07437e9 −0.971300 −0.485650 0.874153i \(-0.661417\pi\)
−0.485650 + 0.874153i \(0.661417\pi\)
\(464\) −2.89715e9 −1.34635
\(465\) 6.26534e8 0.288975
\(466\) −2.11466e9 −0.968031
\(467\) −1.86804e9 −0.848743 −0.424372 0.905488i \(-0.639505\pi\)
−0.424372 + 0.905488i \(0.639505\pi\)
\(468\) −9.30305e8 −0.419532
\(469\) 2.46409e8 0.110294
\(470\) 3.07014e9 1.36401
\(471\) −1.03753e9 −0.457538
\(472\) 4.76973e7 0.0208784
\(473\) −3.45281e8 −0.150023
\(474\) 1.39771e9 0.602826
\(475\) 1.27307e9 0.545034
\(476\) −3.73710e7 −0.0158822
\(477\) 1.08989e9 0.459798
\(478\) −2.06184e9 −0.863491
\(479\) 7.39665e8 0.307511 0.153755 0.988109i \(-0.450863\pi\)
0.153755 + 0.988109i \(0.450863\pi\)
\(480\) 9.60883e8 0.396576
\(481\) 1.48008e9 0.606426
\(482\) −2.83980e9 −1.15511
\(483\) 8.12329e7 0.0328033
\(484\) −2.16801e9 −0.869166
\(485\) −1.28358e9 −0.510890
\(486\) 2.22774e8 0.0880314
\(487\) −3.16893e9 −1.24326 −0.621629 0.783312i \(-0.713529\pi\)
−0.621629 + 0.783312i \(0.713529\pi\)
\(488\) 5.61763e8 0.218818
\(489\) −1.26838e9 −0.490532
\(490\) 1.78920e9 0.687027
\(491\) −1.78015e9 −0.678692 −0.339346 0.940662i \(-0.610206\pi\)
−0.339346 + 0.940662i \(0.610206\pi\)
\(492\) −8.45685e8 −0.320133
\(493\) −5.03295e8 −0.189173
\(494\) 3.84673e9 1.43565
\(495\) 5.74113e7 0.0212755
\(496\) 2.95696e9 1.08808
\(497\) 5.55471e6 0.00202962
\(498\) −2.38585e9 −0.865648
\(499\) −1.30086e9 −0.468684 −0.234342 0.972154i \(-0.575294\pi\)
−0.234342 + 0.972154i \(0.575294\pi\)
\(500\) 2.18275e9 0.780926
\(501\) 1.41352e9 0.502191
\(502\) −5.72937e9 −2.02136
\(503\) −1.92822e9 −0.675569 −0.337784 0.941224i \(-0.609677\pi\)
−0.337784 + 0.941224i \(0.609677\pi\)
\(504\) −1.78252e7 −0.00620192
\(505\) −2.48107e8 −0.0857274
\(506\) −2.46322e8 −0.0845232
\(507\) 1.74679e9 0.595268
\(508\) 1.57192e8 0.0531996
\(509\) −3.91015e9 −1.31426 −0.657129 0.753778i \(-0.728229\pi\)
−0.657129 + 0.753778i \(0.728229\pi\)
\(510\) 1.86703e8 0.0623241
\(511\) 1.18484e8 0.0392815
\(512\) 3.99762e9 1.31631
\(513\) −4.31993e8 −0.141275
\(514\) −3.29120e9 −1.06902
\(515\) −2.39469e9 −0.772546
\(516\) 1.89809e9 0.608195
\(517\) 7.74027e8 0.246343
\(518\) −2.14308e8 −0.0677462
\(519\) −2.84682e9 −0.893870
\(520\) −3.71887e8 −0.115984
\(521\) 3.14164e9 0.973251 0.486626 0.873611i \(-0.338228\pi\)
0.486626 + 0.873611i \(0.338228\pi\)
\(522\) 1.81413e9 0.558240
\(523\) −3.32735e8 −0.101705 −0.0508525 0.998706i \(-0.516194\pi\)
−0.0508525 + 0.998706i \(0.516194\pi\)
\(524\) −4.70627e8 −0.142895
\(525\) −1.64891e8 −0.0497325
\(526\) −1.96712e9 −0.589359
\(527\) 5.13684e8 0.152883
\(528\) 2.70955e8 0.0801085
\(529\) −2.58824e9 −0.760169
\(530\) −3.29240e9 −0.960610
\(531\) −1.49721e8 −0.0433963
\(532\) −2.61211e8 −0.0752143
\(533\) 3.12801e9 0.894793
\(534\) −2.10475e9 −0.598145
\(535\) −1.09137e9 −0.308129
\(536\) −5.43533e8 −0.152458
\(537\) −2.09039e9 −0.582529
\(538\) −4.15185e9 −1.14949
\(539\) 4.51084e8 0.124079
\(540\) −3.15604e8 −0.0862510
\(541\) 3.91262e9 1.06237 0.531187 0.847255i \(-0.321746\pi\)
0.531187 + 0.847255i \(0.321746\pi\)
\(542\) −3.29476e8 −0.0888847
\(543\) 3.89838e9 1.04492
\(544\) 7.87811e8 0.209810
\(545\) 3.28571e9 0.869443
\(546\) −4.98240e8 −0.130998
\(547\) −2.61916e9 −0.684237 −0.342118 0.939657i \(-0.611144\pi\)
−0.342118 + 0.939657i \(0.611144\pi\)
\(548\) 4.20038e9 1.09033
\(549\) −1.76337e9 −0.454820
\(550\) 4.99998e8 0.128144
\(551\) −3.51787e9 −0.895879
\(552\) −1.79185e8 −0.0453434
\(553\) 3.51055e8 0.0882749
\(554\) 4.53318e9 1.13271
\(555\) 5.02114e8 0.124674
\(556\) −8.35131e7 −0.0206060
\(557\) −4.99537e9 −1.22483 −0.612413 0.790538i \(-0.709801\pi\)
−0.612413 + 0.790538i \(0.709801\pi\)
\(558\) −1.85157e9 −0.451151
\(559\) −7.02063e9 −1.69994
\(560\) 2.69935e8 0.0649532
\(561\) 4.70705e7 0.0112559
\(562\) 4.03408e9 0.958666
\(563\) 1.65832e9 0.391643 0.195821 0.980640i \(-0.437263\pi\)
0.195821 + 0.980640i \(0.437263\pi\)
\(564\) −4.25501e9 −0.998674
\(565\) −2.86137e9 −0.667427
\(566\) 5.61292e9 1.30116
\(567\) 5.59530e7 0.0128909
\(568\) −1.22527e7 −0.00280551
\(569\) 6.06474e9 1.38013 0.690064 0.723748i \(-0.257582\pi\)
0.690064 + 0.723748i \(0.257582\pi\)
\(570\) 1.30499e9 0.295152
\(571\) 7.56971e9 1.70158 0.850791 0.525504i \(-0.176123\pi\)
0.850791 + 0.525504i \(0.176123\pi\)
\(572\) 7.08524e8 0.158295
\(573\) 4.94621e9 1.09833
\(574\) −4.52920e8 −0.0999608
\(575\) −1.65755e9 −0.363604
\(576\) −1.15305e9 −0.251403
\(577\) 4.65513e9 1.00883 0.504413 0.863462i \(-0.331709\pi\)
0.504413 + 0.863462i \(0.331709\pi\)
\(578\) −6.21764e9 −1.33930
\(579\) 2.44996e9 0.524547
\(580\) −2.57007e9 −0.546949
\(581\) −5.99243e8 −0.126761
\(582\) 3.79332e9 0.797608
\(583\) −8.30062e8 −0.173488
\(584\) −2.61354e8 −0.0542981
\(585\) 1.16735e9 0.241077
\(586\) −1.01250e10 −2.07852
\(587\) −4.59010e9 −0.936675 −0.468337 0.883550i \(-0.655147\pi\)
−0.468337 + 0.883550i \(0.655147\pi\)
\(588\) −2.47972e9 −0.503016
\(589\) 3.59049e9 0.724019
\(590\) 4.52288e8 0.0906636
\(591\) 2.50364e9 0.498903
\(592\) 2.36975e9 0.469435
\(593\) −3.99474e9 −0.786679 −0.393339 0.919393i \(-0.628680\pi\)
−0.393339 + 0.919393i \(0.628680\pi\)
\(594\) −1.69666e8 −0.0332155
\(595\) 4.68932e7 0.00912643
\(596\) −8.09650e9 −1.56652
\(597\) 2.14586e9 0.412753
\(598\) −5.00848e9 −0.957750
\(599\) −7.94405e9 −1.51025 −0.755124 0.655582i \(-0.772423\pi\)
−0.755124 + 0.655582i \(0.772423\pi\)
\(600\) 3.63720e8 0.0687445
\(601\) 2.34327e9 0.440313 0.220157 0.975465i \(-0.429343\pi\)
0.220157 + 0.975465i \(0.429343\pi\)
\(602\) 1.01655e9 0.189907
\(603\) 1.70615e9 0.316888
\(604\) −3.95595e9 −0.730502
\(605\) 2.72043e9 0.499452
\(606\) 7.33222e8 0.133839
\(607\) 4.14140e9 0.751600 0.375800 0.926701i \(-0.377368\pi\)
0.375800 + 0.926701i \(0.377368\pi\)
\(608\) 5.50654e9 0.993611
\(609\) 4.55645e8 0.0817459
\(610\) 5.32690e9 0.950211
\(611\) 1.57384e10 2.79136
\(612\) −2.58758e8 −0.0456314
\(613\) −8.67426e9 −1.52097 −0.760485 0.649355i \(-0.775039\pi\)
−0.760485 + 0.649355i \(0.775039\pi\)
\(614\) −8.30337e9 −1.44766
\(615\) 1.06117e9 0.183959
\(616\) 1.35757e7 0.00234008
\(617\) 6.63163e9 1.13664 0.568319 0.822808i \(-0.307594\pi\)
0.568319 + 0.822808i \(0.307594\pi\)
\(618\) 7.07694e9 1.20611
\(619\) 1.16526e9 0.197473 0.0987364 0.995114i \(-0.468520\pi\)
0.0987364 + 0.995114i \(0.468520\pi\)
\(620\) 2.62312e9 0.442026
\(621\) 5.62460e8 0.0942477
\(622\) −1.24782e10 −2.07914
\(623\) −5.28639e8 −0.0875893
\(624\) 5.50936e9 0.907726
\(625\) 1.79272e9 0.293719
\(626\) −5.70376e9 −0.929290
\(627\) 3.29008e8 0.0533052
\(628\) −4.34384e9 −0.699866
\(629\) 4.11674e8 0.0659593
\(630\) −1.69027e8 −0.0269317
\(631\) −5.61888e9 −0.890322 −0.445161 0.895450i \(-0.646854\pi\)
−0.445161 + 0.895450i \(0.646854\pi\)
\(632\) −7.74362e8 −0.122021
\(633\) 3.42118e9 0.536121
\(634\) −1.05326e10 −1.64144
\(635\) −1.97246e8 −0.0305703
\(636\) 4.56304e9 0.703323
\(637\) 9.17194e9 1.40596
\(638\) −1.38165e9 −0.210632
\(639\) 3.84610e7 0.00583133
\(640\) −1.07207e9 −0.161657
\(641\) 9.83028e8 0.147422 0.0737111 0.997280i \(-0.476516\pi\)
0.0737111 + 0.997280i \(0.476516\pi\)
\(642\) 3.22528e9 0.481055
\(643\) 7.69653e9 1.14171 0.570856 0.821050i \(-0.306612\pi\)
0.570856 + 0.821050i \(0.306612\pi\)
\(644\) 3.40099e8 0.0501771
\(645\) −2.38173e9 −0.349489
\(646\) 1.06994e9 0.156151
\(647\) −4.42661e9 −0.642549 −0.321275 0.946986i \(-0.604111\pi\)
−0.321275 + 0.946986i \(0.604111\pi\)
\(648\) −1.23422e8 −0.0178189
\(649\) 1.14028e8 0.0163741
\(650\) 1.01665e10 1.45203
\(651\) −4.65050e8 −0.0660642
\(652\) −5.31034e9 −0.750336
\(653\) −3.99185e7 −0.00561020 −0.00280510 0.999996i \(-0.500893\pi\)
−0.00280510 + 0.999996i \(0.500893\pi\)
\(654\) −9.71014e9 −1.35739
\(655\) 5.90545e8 0.0821124
\(656\) 5.00823e9 0.692661
\(657\) 8.20389e8 0.112860
\(658\) −2.27884e9 −0.311834
\(659\) 2.06680e9 0.281319 0.140660 0.990058i \(-0.455078\pi\)
0.140660 + 0.990058i \(0.455078\pi\)
\(660\) 2.40365e8 0.0325437
\(661\) −5.72369e8 −0.0770853 −0.0385426 0.999257i \(-0.512272\pi\)
−0.0385426 + 0.999257i \(0.512272\pi\)
\(662\) −2.79694e9 −0.374697
\(663\) 9.57090e8 0.127543
\(664\) 1.32182e9 0.175220
\(665\) 3.27768e8 0.0432207
\(666\) −1.48388e9 −0.194643
\(667\) 4.58030e9 0.597660
\(668\) 5.91800e9 0.768170
\(669\) 2.92605e9 0.377824
\(670\) −5.15404e9 −0.662043
\(671\) 1.34299e9 0.171610
\(672\) −7.13224e8 −0.0906637
\(673\) −5.59964e9 −0.708121 −0.354060 0.935223i \(-0.615199\pi\)
−0.354060 + 0.935223i \(0.615199\pi\)
\(674\) −9.48014e9 −1.19263
\(675\) −1.14171e9 −0.142887
\(676\) 7.31331e9 0.910543
\(677\) 2.66208e9 0.329731 0.164866 0.986316i \(-0.447281\pi\)
0.164866 + 0.986316i \(0.447281\pi\)
\(678\) 8.45609e9 1.04199
\(679\) 9.52749e8 0.116798
\(680\) −1.03438e8 −0.0126153
\(681\) 4.93001e9 0.598181
\(682\) 1.41017e9 0.170226
\(683\) −3.08492e9 −0.370485 −0.185243 0.982693i \(-0.559307\pi\)
−0.185243 + 0.982693i \(0.559307\pi\)
\(684\) −1.80863e9 −0.216100
\(685\) −5.27065e9 −0.626538
\(686\) −2.67422e9 −0.316274
\(687\) 4.35348e9 0.512257
\(688\) −1.12407e10 −1.31593
\(689\) −1.68777e10 −1.96583
\(690\) −1.69912e9 −0.196902
\(691\) 1.50276e9 0.173267 0.0866336 0.996240i \(-0.472389\pi\)
0.0866336 + 0.996240i \(0.472389\pi\)
\(692\) −1.19188e10 −1.36730
\(693\) −4.26140e7 −0.00486392
\(694\) −3.07152e9 −0.348815
\(695\) 1.04793e8 0.0118409
\(696\) −1.00507e9 −0.112996
\(697\) 8.70033e8 0.0973243
\(698\) −1.40022e10 −1.55849
\(699\) −3.67754e9 −0.407275
\(700\) −6.90354e8 −0.0760726
\(701\) −8.60128e9 −0.943083 −0.471541 0.881844i \(-0.656302\pi\)
−0.471541 + 0.881844i \(0.656302\pi\)
\(702\) −3.44983e9 −0.376372
\(703\) 2.87747e9 0.312368
\(704\) 8.78171e8 0.0948582
\(705\) 5.33920e9 0.573871
\(706\) −1.33251e10 −1.42512
\(707\) 1.84160e8 0.0195987
\(708\) −6.26841e8 −0.0663805
\(709\) 1.22113e10 1.28677 0.643384 0.765544i \(-0.277530\pi\)
0.643384 + 0.765544i \(0.277530\pi\)
\(710\) −1.16186e8 −0.0121828
\(711\) 2.43072e9 0.253624
\(712\) 1.16608e9 0.121073
\(713\) −4.67485e9 −0.483008
\(714\) −1.38582e8 −0.0142483
\(715\) −8.89059e8 −0.0909619
\(716\) −8.75188e9 −0.891057
\(717\) −3.58570e9 −0.363292
\(718\) −7.38882e9 −0.744972
\(719\) 3.71959e9 0.373202 0.186601 0.982436i \(-0.440253\pi\)
0.186601 + 0.982436i \(0.440253\pi\)
\(720\) 1.86904e9 0.186618
\(721\) 1.77748e9 0.176616
\(722\) −6.39927e9 −0.632777
\(723\) −4.93862e9 −0.485984
\(724\) 1.63214e10 1.59835
\(725\) −9.29737e9 −0.906102
\(726\) −8.03958e9 −0.779750
\(727\) 1.15890e10 1.11861 0.559303 0.828963i \(-0.311069\pi\)
0.559303 + 0.828963i \(0.311069\pi\)
\(728\) 2.76036e8 0.0265159
\(729\) 3.87420e8 0.0370370
\(730\) −2.47829e9 −0.235788
\(731\) −1.95274e9 −0.184898
\(732\) −7.38273e9 −0.695710
\(733\) 1.55996e10 1.46302 0.731510 0.681830i \(-0.238816\pi\)
0.731510 + 0.681830i \(0.238816\pi\)
\(734\) 1.11450e10 1.04027
\(735\) 3.11156e9 0.289050
\(736\) −7.16958e9 −0.662859
\(737\) −1.29941e9 −0.119566
\(738\) −3.13603e9 −0.287199
\(739\) −1.06655e10 −0.972137 −0.486069 0.873921i \(-0.661569\pi\)
−0.486069 + 0.873921i \(0.661569\pi\)
\(740\) 2.10221e9 0.190706
\(741\) 6.68975e9 0.604012
\(742\) 2.44381e9 0.219611
\(743\) −7.49599e9 −0.670453 −0.335226 0.942138i \(-0.608813\pi\)
−0.335226 + 0.942138i \(0.608813\pi\)
\(744\) 1.02581e9 0.0913195
\(745\) 1.01595e10 0.900173
\(746\) −9.09743e9 −0.802292
\(747\) −4.14918e9 −0.364200
\(748\) 1.97071e8 0.0172174
\(749\) 8.10077e8 0.0704433
\(750\) 8.09426e9 0.700588
\(751\) 1.56605e10 1.34916 0.674582 0.738200i \(-0.264324\pi\)
0.674582 + 0.738200i \(0.264324\pi\)
\(752\) 2.51986e10 2.16080
\(753\) −9.96379e9 −0.850437
\(754\) −2.80932e10 −2.38672
\(755\) 4.96394e9 0.419771
\(756\) 2.34259e8 0.0197184
\(757\) 2.02856e10 1.69962 0.849810 0.527090i \(-0.176717\pi\)
0.849810 + 0.527090i \(0.176717\pi\)
\(758\) 2.70857e9 0.225890
\(759\) −4.28371e8 −0.0355610
\(760\) −7.22997e8 −0.0597432
\(761\) −1.72538e10 −1.41919 −0.709594 0.704611i \(-0.751121\pi\)
−0.709594 + 0.704611i \(0.751121\pi\)
\(762\) 5.82913e8 0.0477267
\(763\) −2.43884e9 −0.198769
\(764\) 2.07084e10 1.68004
\(765\) 3.24690e8 0.0262213
\(766\) −9.70235e9 −0.779967
\(767\) 2.31855e9 0.185538
\(768\) 8.63460e9 0.687825
\(769\) 2.62229e9 0.207940 0.103970 0.994580i \(-0.466845\pi\)
0.103970 + 0.994580i \(0.466845\pi\)
\(770\) 1.28731e8 0.0101617
\(771\) −5.72365e9 −0.449762
\(772\) 1.02573e10 0.802366
\(773\) −1.36251e10 −1.06099 −0.530497 0.847687i \(-0.677995\pi\)
−0.530497 + 0.847687i \(0.677995\pi\)
\(774\) 7.03864e9 0.545627
\(775\) 9.48929e9 0.732281
\(776\) −2.10159e9 −0.161448
\(777\) −3.72698e8 −0.0285025
\(778\) −3.18280e9 −0.242315
\(779\) 6.08125e9 0.460905
\(780\) 4.88737e9 0.368760
\(781\) −2.92921e7 −0.00220025
\(782\) −1.39307e9 −0.104172
\(783\) 3.15490e9 0.234866
\(784\) 1.46851e10 1.08836
\(785\) 5.45067e9 0.402166
\(786\) −1.74522e9 −0.128195
\(787\) −1.27830e9 −0.0934803 −0.0467402 0.998907i \(-0.514883\pi\)
−0.0467402 + 0.998907i \(0.514883\pi\)
\(788\) 1.04820e10 0.763140
\(789\) −3.42096e9 −0.247958
\(790\) −7.34287e9 −0.529872
\(791\) 2.12387e9 0.152585
\(792\) 9.39987e7 0.00672332
\(793\) 2.73071e10 1.94455
\(794\) −3.05403e10 −2.16522
\(795\) −5.72573e9 −0.404153
\(796\) 8.98409e9 0.631362
\(797\) −1.05141e10 −0.735642 −0.367821 0.929897i \(-0.619896\pi\)
−0.367821 + 0.929897i \(0.619896\pi\)
\(798\) −9.68642e8 −0.0674766
\(799\) 4.37752e9 0.303609
\(800\) 1.45532e10 1.00495
\(801\) −3.66032e9 −0.251654
\(802\) −3.24754e10 −2.22303
\(803\) −6.24812e8 −0.0425838
\(804\) 7.14315e9 0.484723
\(805\) −4.26758e8 −0.0288334
\(806\) 2.86731e10 1.92886
\(807\) −7.22037e9 −0.483617
\(808\) −4.06222e8 −0.0270909
\(809\) −6.76915e7 −0.00449484 −0.00224742 0.999997i \(-0.500715\pi\)
−0.00224742 + 0.999997i \(0.500715\pi\)
\(810\) −1.17035e9 −0.0773778
\(811\) −1.33239e10 −0.877121 −0.438560 0.898702i \(-0.644511\pi\)
−0.438560 + 0.898702i \(0.644511\pi\)
\(812\) 1.90766e9 0.125041
\(813\) −5.72984e8 −0.0373960
\(814\) 1.13013e9 0.0734416
\(815\) 6.66343e9 0.431168
\(816\) 1.53239e9 0.0987310
\(817\) −1.36490e10 −0.875637
\(818\) 5.69488e9 0.363788
\(819\) −8.66475e8 −0.0551141
\(820\) 4.44281e9 0.281391
\(821\) 1.93379e10 1.21957 0.609787 0.792565i \(-0.291255\pi\)
0.609787 + 0.792565i \(0.291255\pi\)
\(822\) 1.55762e10 0.978159
\(823\) −1.01992e10 −0.637774 −0.318887 0.947793i \(-0.603309\pi\)
−0.318887 + 0.947793i \(0.603309\pi\)
\(824\) −3.92079e9 −0.244134
\(825\) 8.69534e8 0.0539135
\(826\) −3.35715e8 −0.0207272
\(827\) −2.20862e9 −0.135785 −0.0678925 0.997693i \(-0.521628\pi\)
−0.0678925 + 0.997693i \(0.521628\pi\)
\(828\) 2.35486e9 0.144165
\(829\) 5.49783e9 0.335159 0.167579 0.985859i \(-0.446405\pi\)
0.167579 + 0.985859i \(0.446405\pi\)
\(830\) 1.25341e10 0.760887
\(831\) 7.88353e9 0.476560
\(832\) 1.78559e10 1.07486
\(833\) 2.55111e9 0.152923
\(834\) −3.09690e8 −0.0184861
\(835\) −7.42593e9 −0.441416
\(836\) 1.37746e9 0.0815376
\(837\) −3.22002e9 −0.189810
\(838\) 1.97647e10 1.16021
\(839\) 4.86845e9 0.284593 0.142297 0.989824i \(-0.454551\pi\)
0.142297 + 0.989824i \(0.454551\pi\)
\(840\) 9.36446e7 0.00545137
\(841\) 8.44158e9 0.489371
\(842\) −1.08413e10 −0.625876
\(843\) 7.01556e9 0.403335
\(844\) 1.43235e10 0.820070
\(845\) −9.17677e9 −0.523229
\(846\) −1.57788e10 −0.895935
\(847\) −2.01926e9 −0.114183
\(848\) −2.70228e10 −1.52176
\(849\) 9.76128e9 0.547432
\(850\) 2.82774e9 0.157933
\(851\) −3.74649e9 −0.208387
\(852\) 1.61025e8 0.00891982
\(853\) 2.22806e10 1.22915 0.614577 0.788857i \(-0.289327\pi\)
0.614577 + 0.788857i \(0.289327\pi\)
\(854\) −3.95394e9 −0.217234
\(855\) 2.26948e9 0.124178
\(856\) −1.78688e9 −0.0973727
\(857\) 3.09258e10 1.67837 0.839186 0.543844i \(-0.183032\pi\)
0.839186 + 0.543844i \(0.183032\pi\)
\(858\) 2.62740e9 0.142011
\(859\) 6.06715e9 0.326594 0.163297 0.986577i \(-0.447787\pi\)
0.163297 + 0.986577i \(0.447787\pi\)
\(860\) −9.97163e9 −0.534591
\(861\) −7.87661e8 −0.0420560
\(862\) 4.09674e9 0.217853
\(863\) −5.47212e8 −0.0289813 −0.0144906 0.999895i \(-0.504613\pi\)
−0.0144906 + 0.999895i \(0.504613\pi\)
\(864\) −4.93838e9 −0.260487
\(865\) 1.49558e10 0.785694
\(866\) 3.32052e10 1.73738
\(867\) −1.08129e10 −0.563478
\(868\) −1.94703e9 −0.101054
\(869\) −1.85124e9 −0.0956961
\(870\) −9.53053e9 −0.490682
\(871\) −2.64210e10 −1.35483
\(872\) 5.37964e9 0.274755
\(873\) 6.59687e9 0.335574
\(874\) −9.73714e9 −0.493334
\(875\) 2.03299e9 0.102591
\(876\) 3.43474e9 0.172635
\(877\) −1.05471e10 −0.528001 −0.264001 0.964523i \(-0.585042\pi\)
−0.264001 + 0.964523i \(0.585042\pi\)
\(878\) 4.35310e10 2.17054
\(879\) −1.76082e10 −0.874487
\(880\) −1.42347e9 −0.0704138
\(881\) −2.80065e10 −1.37989 −0.689943 0.723864i \(-0.742364\pi\)
−0.689943 + 0.723864i \(0.742364\pi\)
\(882\) −9.19548e9 −0.451268
\(883\) −1.61306e10 −0.788477 −0.394238 0.919008i \(-0.628992\pi\)
−0.394238 + 0.919008i \(0.628992\pi\)
\(884\) 4.00706e9 0.195094
\(885\) 7.86562e8 0.0381445
\(886\) 3.89940e10 1.88356
\(887\) 8.47554e9 0.407788 0.203894 0.978993i \(-0.434640\pi\)
0.203894 + 0.978993i \(0.434640\pi\)
\(888\) 8.22103e8 0.0393986
\(889\) 1.46407e8 0.00698886
\(890\) 1.10573e10 0.525757
\(891\) −2.95061e8 −0.0139746
\(892\) 1.22505e10 0.577933
\(893\) 3.05974e10 1.43782
\(894\) −3.00241e10 −1.40536
\(895\) 1.09819e10 0.512031
\(896\) 7.95756e8 0.0369574
\(897\) −8.71012e9 −0.402949
\(898\) 4.55357e10 2.09838
\(899\) −2.62218e10 −1.20366
\(900\) −4.78003e9 −0.218566
\(901\) −4.69442e9 −0.213818
\(902\) 2.38842e9 0.108365
\(903\) 1.76786e9 0.0798989
\(904\) −4.68487e9 −0.210915
\(905\) −2.04802e10 −0.918466
\(906\) −1.46697e10 −0.655351
\(907\) 1.59071e10 0.707892 0.353946 0.935266i \(-0.384840\pi\)
0.353946 + 0.935266i \(0.384840\pi\)
\(908\) 2.06406e10 0.914999
\(909\) 1.27513e9 0.0563093
\(910\) 2.61751e9 0.115144
\(911\) −7.44930e9 −0.326438 −0.163219 0.986590i \(-0.552188\pi\)
−0.163219 + 0.986590i \(0.552188\pi\)
\(912\) 1.07109e10 0.467567
\(913\) 3.16003e9 0.137418
\(914\) −5.12923e10 −2.22198
\(915\) 9.26387e9 0.399778
\(916\) 1.82268e10 0.783567
\(917\) −4.38337e8 −0.0187722
\(918\) −9.59545e8 −0.0409370
\(919\) −2.63715e10 −1.12080 −0.560402 0.828221i \(-0.689353\pi\)
−0.560402 + 0.828221i \(0.689353\pi\)
\(920\) 9.41349e8 0.0398560
\(921\) −1.44402e10 −0.609065
\(922\) 9.75990e8 0.0410098
\(923\) −5.95599e8 −0.0249315
\(924\) −1.78413e8 −0.00744003
\(925\) 7.60485e9 0.315933
\(926\) −3.22057e10 −1.33289
\(927\) 1.23073e10 0.507440
\(928\) −4.02150e10 −1.65185
\(929\) 1.48686e9 0.0608437 0.0304219 0.999537i \(-0.490315\pi\)
0.0304219 + 0.999537i \(0.490315\pi\)
\(930\) 9.72726e9 0.396552
\(931\) 1.78314e10 0.724207
\(932\) −1.53968e10 −0.622983
\(933\) −2.17004e10 −0.874748
\(934\) −2.90022e10 −1.16471
\(935\) −2.47285e8 −0.00989368
\(936\) 1.91128e9 0.0761833
\(937\) 1.58188e10 0.628182 0.314091 0.949393i \(-0.398300\pi\)
0.314091 + 0.949393i \(0.398300\pi\)
\(938\) 3.82563e9 0.151354
\(939\) −9.91926e9 −0.390976
\(940\) 2.23537e10 0.877814
\(941\) −7.02941e9 −0.275014 −0.137507 0.990501i \(-0.543909\pi\)
−0.137507 + 0.990501i \(0.543909\pi\)
\(942\) −1.61082e10 −0.627867
\(943\) −7.91785e9 −0.307480
\(944\) 3.71221e9 0.143625
\(945\) −2.93950e8 −0.0113308
\(946\) −5.36066e9 −0.205873
\(947\) 4.03890e10 1.54539 0.772695 0.634778i \(-0.218908\pi\)
0.772695 + 0.634778i \(0.218908\pi\)
\(948\) 1.01767e10 0.387953
\(949\) −1.27044e10 −0.482526
\(950\) 1.97650e10 0.747936
\(951\) −1.83170e10 −0.690593
\(952\) 7.67775e7 0.00288406
\(953\) 2.02572e10 0.758148 0.379074 0.925366i \(-0.376243\pi\)
0.379074 + 0.925366i \(0.376243\pi\)
\(954\) 1.69210e10 0.630968
\(955\) −2.59850e10 −0.965406
\(956\) −1.50123e10 −0.555705
\(957\) −2.40279e9 −0.0886182
\(958\) 1.14837e10 0.421989
\(959\) 3.91219e9 0.143237
\(960\) 6.05758e9 0.220979
\(961\) −7.49560e8 −0.0272442
\(962\) 2.29790e10 0.832182
\(963\) 5.60900e9 0.202392
\(964\) −2.06766e10 −0.743378
\(965\) −1.28709e10 −0.461066
\(966\) 1.26118e9 0.0450151
\(967\) 1.14238e10 0.406273 0.203137 0.979150i \(-0.434886\pi\)
0.203137 + 0.979150i \(0.434886\pi\)
\(968\) 4.45411e9 0.157833
\(969\) 1.86071e9 0.0656968
\(970\) −1.99283e10 −0.701081
\(971\) 2.91162e10 1.02063 0.510314 0.859988i \(-0.329529\pi\)
0.510314 + 0.859988i \(0.329529\pi\)
\(972\) 1.62202e9 0.0566532
\(973\) −7.77832e7 −0.00270701
\(974\) −4.91992e10 −1.70609
\(975\) 1.76803e10 0.610905
\(976\) 4.37212e10 1.50528
\(977\) 4.46068e10 1.53028 0.765139 0.643865i \(-0.222670\pi\)
0.765139 + 0.643865i \(0.222670\pi\)
\(978\) −1.96922e10 −0.673145
\(979\) 2.78771e9 0.0949529
\(980\) 1.30272e10 0.442141
\(981\) −1.68866e10 −0.571086
\(982\) −2.76378e10 −0.931351
\(983\) −6.33957e9 −0.212874 −0.106437 0.994319i \(-0.533944\pi\)
−0.106437 + 0.994319i \(0.533944\pi\)
\(984\) 1.73744e9 0.0581334
\(985\) −1.31529e10 −0.438526
\(986\) −7.81391e9 −0.259597
\(987\) −3.96307e9 −0.131196
\(988\) 2.80081e10 0.923919
\(989\) 1.77712e10 0.584156
\(990\) 8.91340e8 0.0291958
\(991\) −2.38700e10 −0.779103 −0.389551 0.921005i \(-0.627370\pi\)
−0.389551 + 0.921005i \(0.627370\pi\)
\(992\) 4.10451e10 1.33497
\(993\) −4.86409e9 −0.157645
\(994\) 8.62397e7 0.00278519
\(995\) −1.12733e10 −0.362802
\(996\) −1.73714e10 −0.557093
\(997\) −1.70894e9 −0.0546127 −0.0273063 0.999627i \(-0.508693\pi\)
−0.0273063 + 0.999627i \(0.508693\pi\)
\(998\) −2.01966e10 −0.643163
\(999\) −2.58057e9 −0.0818911
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.b.1.15 17
3.2 odd 2 531.8.a.d.1.3 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.15 17 1.1 even 1 trivial
531.8.a.d.1.3 17 3.2 odd 2