Properties

Label 177.8.a.b.1.4
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
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 1.4
Root \(-15.8998\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-17.8998 q^{2} +27.0000 q^{3} +192.403 q^{4} +255.355 q^{5} -483.295 q^{6} -1072.82 q^{7} -1152.80 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-17.8998 q^{2} +27.0000 q^{3} +192.403 q^{4} +255.355 q^{5} -483.295 q^{6} -1072.82 q^{7} -1152.80 q^{8} +729.000 q^{9} -4570.81 q^{10} +4742.70 q^{11} +5194.88 q^{12} +5721.05 q^{13} +19203.3 q^{14} +6894.59 q^{15} -3992.71 q^{16} -12850.6 q^{17} -13049.0 q^{18} -52123.1 q^{19} +49131.1 q^{20} -28966.2 q^{21} -84893.4 q^{22} -9824.17 q^{23} -31125.6 q^{24} -12918.7 q^{25} -102406. q^{26} +19683.0 q^{27} -206414. q^{28} -196436. q^{29} -123412. q^{30} +272544. q^{31} +219027. q^{32} +128053. q^{33} +230023. q^{34} -273951. q^{35} +140262. q^{36} +309341. q^{37} +932994. q^{38} +154468. q^{39} -294373. q^{40} -421229. q^{41} +518490. q^{42} -406292. q^{43} +912509. q^{44} +186154. q^{45} +175851. q^{46} -261121. q^{47} -107803. q^{48} +327408. q^{49} +231242. q^{50} -346967. q^{51} +1.10075e6 q^{52} +738907. q^{53} -352322. q^{54} +1.21107e6 q^{55} +1.23675e6 q^{56} -1.40732e6 q^{57} +3.51616e6 q^{58} -205379. q^{59} +1.32654e6 q^{60} +233563. q^{61} -4.87848e6 q^{62} -782089. q^{63} -3.40947e6 q^{64} +1.46090e6 q^{65} -2.29212e6 q^{66} +3.54145e6 q^{67} -2.47250e6 q^{68} -265253. q^{69} +4.90367e6 q^{70} +1.03036e6 q^{71} -840390. q^{72} -5.88757e6 q^{73} -5.53715e6 q^{74} -348804. q^{75} -1.00286e7 q^{76} -5.08808e6 q^{77} -2.76495e6 q^{78} +4.34224e6 q^{79} -1.01956e6 q^{80} +531441. q^{81} +7.53991e6 q^{82} -4.12014e6 q^{83} -5.57319e6 q^{84} -3.28147e6 q^{85} +7.27255e6 q^{86} -5.30376e6 q^{87} -5.46737e6 q^{88} -7.32690e6 q^{89} -3.33212e6 q^{90} -6.13768e6 q^{91} -1.89020e6 q^{92} +7.35868e6 q^{93} +4.67402e6 q^{94} -1.33099e7 q^{95} +5.91372e6 q^{96} +1.46546e7 q^{97} -5.86054e6 q^{98} +3.45743e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −17.8998 −1.58213 −0.791067 0.611730i \(-0.790474\pi\)
−0.791067 + 0.611730i \(0.790474\pi\)
\(3\) 27.0000 0.577350
\(4\) 192.403 1.50315
\(5\) 255.355 0.913587 0.456793 0.889573i \(-0.348998\pi\)
0.456793 + 0.889573i \(0.348998\pi\)
\(6\) −483.295 −0.913445
\(7\) −1072.82 −1.18218 −0.591092 0.806604i \(-0.701303\pi\)
−0.591092 + 0.806604i \(0.701303\pi\)
\(8\) −1152.80 −0.796046
\(9\) 729.000 0.333333
\(10\) −4570.81 −1.44542
\(11\) 4742.70 1.07436 0.537182 0.843467i \(-0.319489\pi\)
0.537182 + 0.843467i \(0.319489\pi\)
\(12\) 5194.88 0.867843
\(13\) 5721.05 0.722227 0.361114 0.932522i \(-0.382397\pi\)
0.361114 + 0.932522i \(0.382397\pi\)
\(14\) 19203.3 1.87037
\(15\) 6894.59 0.527460
\(16\) −3992.71 −0.243695
\(17\) −12850.6 −0.634385 −0.317192 0.948361i \(-0.602740\pi\)
−0.317192 + 0.948361i \(0.602740\pi\)
\(18\) −13049.0 −0.527378
\(19\) −52123.1 −1.74338 −0.871692 0.490055i \(-0.836977\pi\)
−0.871692 + 0.490055i \(0.836977\pi\)
\(20\) 49131.1 1.37326
\(21\) −28966.2 −0.682535
\(22\) −84893.4 −1.69979
\(23\) −9824.17 −0.168364 −0.0841818 0.996450i \(-0.526828\pi\)
−0.0841818 + 0.996450i \(0.526828\pi\)
\(24\) −31125.6 −0.459598
\(25\) −12918.7 −0.165359
\(26\) −102406. −1.14266
\(27\) 19683.0 0.192450
\(28\) −206414. −1.77700
\(29\) −196436. −1.49564 −0.747821 0.663901i \(-0.768900\pi\)
−0.747821 + 0.663901i \(0.768900\pi\)
\(30\) −123412. −0.834512
\(31\) 272544. 1.64312 0.821562 0.570119i \(-0.193103\pi\)
0.821562 + 0.570119i \(0.193103\pi\)
\(32\) 219027. 1.18161
\(33\) 128053. 0.620284
\(34\) 230023. 1.00368
\(35\) −273951. −1.08003
\(36\) 140262. 0.501049
\(37\) 309341. 1.00400 0.501998 0.864869i \(-0.332599\pi\)
0.501998 + 0.864869i \(0.332599\pi\)
\(38\) 932994. 2.75827
\(39\) 154468. 0.416978
\(40\) −294373. −0.727258
\(41\) −421229. −0.954497 −0.477249 0.878768i \(-0.658366\pi\)
−0.477249 + 0.878768i \(0.658366\pi\)
\(42\) 518490. 1.07986
\(43\) −406292. −0.779289 −0.389645 0.920965i \(-0.627402\pi\)
−0.389645 + 0.920965i \(0.627402\pi\)
\(44\) 912509. 1.61493
\(45\) 186154. 0.304529
\(46\) 175851. 0.266374
\(47\) −261121. −0.366859 −0.183430 0.983033i \(-0.558720\pi\)
−0.183430 + 0.983033i \(0.558720\pi\)
\(48\) −107803. −0.140698
\(49\) 327408. 0.397560
\(50\) 231242. 0.261620
\(51\) −346967. −0.366262
\(52\) 1.10075e6 1.08561
\(53\) 738907. 0.681748 0.340874 0.940109i \(-0.389277\pi\)
0.340874 + 0.940109i \(0.389277\pi\)
\(54\) −352322. −0.304482
\(55\) 1.21107e6 0.981524
\(56\) 1.23675e6 0.941074
\(57\) −1.40732e6 −1.00654
\(58\) 3.51616e6 2.36630
\(59\) −205379. −0.130189
\(60\) 1.32654e6 0.792850
\(61\) 233563. 0.131750 0.0658748 0.997828i \(-0.479016\pi\)
0.0658748 + 0.997828i \(0.479016\pi\)
\(62\) −4.87848e6 −2.59964
\(63\) −782089. −0.394062
\(64\) −3.40947e6 −1.62576
\(65\) 1.46090e6 0.659817
\(66\) −2.29212e6 −0.981372
\(67\) 3.54145e6 1.43853 0.719265 0.694736i \(-0.244479\pi\)
0.719265 + 0.694736i \(0.244479\pi\)
\(68\) −2.47250e6 −0.953574
\(69\) −265253. −0.0972048
\(70\) 4.90367e6 1.70875
\(71\) 1.03036e6 0.341652 0.170826 0.985301i \(-0.445356\pi\)
0.170826 + 0.985301i \(0.445356\pi\)
\(72\) −840390. −0.265349
\(73\) −5.88757e6 −1.77136 −0.885679 0.464298i \(-0.846307\pi\)
−0.885679 + 0.464298i \(0.846307\pi\)
\(74\) −5.53715e6 −1.58845
\(75\) −348804. −0.0954700
\(76\) −1.00286e7 −2.62056
\(77\) −5.08808e6 −1.27010
\(78\) −2.76495e6 −0.659715
\(79\) 4.34224e6 0.990875 0.495437 0.868644i \(-0.335008\pi\)
0.495437 + 0.868644i \(0.335008\pi\)
\(80\) −1.01956e6 −0.222637
\(81\) 531441. 0.111111
\(82\) 7.53991e6 1.51014
\(83\) −4.12014e6 −0.790931 −0.395465 0.918481i \(-0.629417\pi\)
−0.395465 + 0.918481i \(0.629417\pi\)
\(84\) −5.57319e6 −1.02595
\(85\) −3.28147e6 −0.579566
\(86\) 7.27255e6 1.23294
\(87\) −5.30376e6 −0.863509
\(88\) −5.46737e6 −0.855243
\(89\) −7.32690e6 −1.10168 −0.550840 0.834611i \(-0.685692\pi\)
−0.550840 + 0.834611i \(0.685692\pi\)
\(90\) −3.33212e6 −0.481806
\(91\) −6.13768e6 −0.853806
\(92\) −1.89020e6 −0.253075
\(93\) 7.35868e6 0.948658
\(94\) 4.67402e6 0.580421
\(95\) −1.33099e7 −1.59273
\(96\) 5.91372e6 0.682200
\(97\) 1.46546e7 1.63032 0.815158 0.579239i \(-0.196650\pi\)
0.815158 + 0.579239i \(0.196650\pi\)
\(98\) −5.86054e6 −0.628994
\(99\) 3.45743e6 0.358121
\(100\) −2.48559e6 −0.248559
\(101\) 3.05329e6 0.294878 0.147439 0.989071i \(-0.452897\pi\)
0.147439 + 0.989071i \(0.452897\pi\)
\(102\) 6.21063e6 0.579476
\(103\) −1.49285e7 −1.34613 −0.673063 0.739585i \(-0.735022\pi\)
−0.673063 + 0.739585i \(0.735022\pi\)
\(104\) −6.59521e6 −0.574926
\(105\) −7.39668e6 −0.623555
\(106\) −1.32263e7 −1.07862
\(107\) −2.22015e7 −1.75202 −0.876011 0.482291i \(-0.839805\pi\)
−0.876011 + 0.482291i \(0.839805\pi\)
\(108\) 3.78707e6 0.289281
\(109\) −2.47303e6 −0.182910 −0.0914549 0.995809i \(-0.529152\pi\)
−0.0914549 + 0.995809i \(0.529152\pi\)
\(110\) −2.16780e7 −1.55290
\(111\) 8.35221e6 0.579657
\(112\) 4.28347e6 0.288093
\(113\) −1.16610e7 −0.760260 −0.380130 0.924933i \(-0.624121\pi\)
−0.380130 + 0.924933i \(0.624121\pi\)
\(114\) 2.51908e7 1.59249
\(115\) −2.50865e6 −0.153815
\(116\) −3.77948e7 −2.24817
\(117\) 4.17064e6 0.240742
\(118\) 3.67624e6 0.205976
\(119\) 1.37864e7 0.749960
\(120\) −7.94808e6 −0.419882
\(121\) 3.00601e6 0.154256
\(122\) −4.18073e6 −0.208446
\(123\) −1.13732e7 −0.551079
\(124\) 5.24382e7 2.46986
\(125\) −2.32485e7 −1.06466
\(126\) 1.39992e7 0.623458
\(127\) −3.47578e7 −1.50570 −0.752852 0.658190i \(-0.771322\pi\)
−0.752852 + 0.658190i \(0.771322\pi\)
\(128\) 3.29934e7 1.39057
\(129\) −1.09699e7 −0.449923
\(130\) −2.61498e7 −1.04392
\(131\) −1.06271e7 −0.413016 −0.206508 0.978445i \(-0.566210\pi\)
−0.206508 + 0.978445i \(0.566210\pi\)
\(132\) 2.46377e7 0.932378
\(133\) 5.59189e7 2.06100
\(134\) −6.33912e7 −2.27595
\(135\) 5.02616e6 0.175820
\(136\) 1.48142e7 0.505000
\(137\) −4.64453e7 −1.54319 −0.771596 0.636112i \(-0.780541\pi\)
−0.771596 + 0.636112i \(0.780541\pi\)
\(138\) 4.74797e6 0.153791
\(139\) −1.75461e7 −0.554151 −0.277076 0.960848i \(-0.589365\pi\)
−0.277076 + 0.960848i \(0.589365\pi\)
\(140\) −5.27090e7 −1.62344
\(141\) −7.05027e6 −0.211806
\(142\) −1.84432e7 −0.540539
\(143\) 2.71332e7 0.775934
\(144\) −2.91068e6 −0.0812318
\(145\) −5.01609e7 −1.36640
\(146\) 1.05386e8 2.80253
\(147\) 8.84002e6 0.229532
\(148\) 5.95181e7 1.50915
\(149\) 6.90672e7 1.71049 0.855244 0.518225i \(-0.173407\pi\)
0.855244 + 0.518225i \(0.173407\pi\)
\(150\) 6.24352e6 0.151046
\(151\) 9.93956e6 0.234935 0.117468 0.993077i \(-0.462522\pi\)
0.117468 + 0.993077i \(0.462522\pi\)
\(152\) 6.00875e7 1.38781
\(153\) −9.36810e6 −0.211462
\(154\) 9.10756e7 2.00946
\(155\) 6.95955e7 1.50114
\(156\) 2.97201e7 0.626780
\(157\) −1.73655e7 −0.358128 −0.179064 0.983837i \(-0.557307\pi\)
−0.179064 + 0.983837i \(0.557307\pi\)
\(158\) −7.77252e7 −1.56770
\(159\) 1.99505e7 0.393608
\(160\) 5.59297e7 1.07950
\(161\) 1.05396e7 0.199037
\(162\) −9.51269e6 −0.175793
\(163\) −8.66389e7 −1.56695 −0.783477 0.621421i \(-0.786556\pi\)
−0.783477 + 0.621421i \(0.786556\pi\)
\(164\) −8.10456e7 −1.43475
\(165\) 3.26990e7 0.566683
\(166\) 7.37497e7 1.25136
\(167\) −2.60451e7 −0.432732 −0.216366 0.976312i \(-0.569420\pi\)
−0.216366 + 0.976312i \(0.569420\pi\)
\(168\) 3.33922e7 0.543329
\(169\) −3.00181e7 −0.478388
\(170\) 5.87377e7 0.916950
\(171\) −3.79978e7 −0.581128
\(172\) −7.81717e7 −1.17139
\(173\) −2.86328e7 −0.420438 −0.210219 0.977654i \(-0.567418\pi\)
−0.210219 + 0.977654i \(0.567418\pi\)
\(174\) 9.49363e7 1.36619
\(175\) 1.38595e7 0.195485
\(176\) −1.89362e7 −0.261817
\(177\) −5.54523e6 −0.0751646
\(178\) 1.31150e8 1.74300
\(179\) 9.04329e6 0.117853 0.0589266 0.998262i \(-0.481232\pi\)
0.0589266 + 0.998262i \(0.481232\pi\)
\(180\) 3.58166e7 0.457752
\(181\) 9.04918e7 1.13432 0.567158 0.823609i \(-0.308043\pi\)
0.567158 + 0.823609i \(0.308043\pi\)
\(182\) 1.09863e8 1.35084
\(183\) 6.30620e6 0.0760657
\(184\) 1.13253e7 0.134025
\(185\) 7.89919e7 0.917237
\(186\) −1.31719e8 −1.50090
\(187\) −6.09466e7 −0.681559
\(188\) −5.02405e7 −0.551444
\(189\) −2.11164e7 −0.227512
\(190\) 2.38245e8 2.51992
\(191\) −7.52927e7 −0.781873 −0.390936 0.920418i \(-0.627849\pi\)
−0.390936 + 0.920418i \(0.627849\pi\)
\(192\) −9.20557e7 −0.938634
\(193\) 3.03390e7 0.303774 0.151887 0.988398i \(-0.451465\pi\)
0.151887 + 0.988398i \(0.451465\pi\)
\(194\) −2.62314e8 −2.57938
\(195\) 3.94443e7 0.380946
\(196\) 6.29942e7 0.597592
\(197\) 5.75348e7 0.536165 0.268083 0.963396i \(-0.413610\pi\)
0.268083 + 0.963396i \(0.413610\pi\)
\(198\) −6.18873e7 −0.566595
\(199\) 3.30510e6 0.0297303 0.0148651 0.999890i \(-0.495268\pi\)
0.0148651 + 0.999890i \(0.495268\pi\)
\(200\) 1.48926e7 0.131633
\(201\) 9.56191e7 0.830536
\(202\) −5.46532e7 −0.466537
\(203\) 2.10741e8 1.76812
\(204\) −6.67574e7 −0.550546
\(205\) −1.07563e8 −0.872016
\(206\) 2.67217e8 2.12975
\(207\) −7.16182e6 −0.0561212
\(208\) −2.28425e7 −0.176003
\(209\) −2.47204e8 −1.87303
\(210\) 1.32399e8 0.986547
\(211\) 7.47470e7 0.547779 0.273890 0.961761i \(-0.411690\pi\)
0.273890 + 0.961761i \(0.411690\pi\)
\(212\) 1.42168e8 1.02477
\(213\) 2.78197e7 0.197253
\(214\) 3.97403e8 2.77193
\(215\) −1.03749e8 −0.711948
\(216\) −2.26905e7 −0.153199
\(217\) −2.92391e8 −1.94248
\(218\) 4.42668e7 0.289388
\(219\) −1.58964e8 −1.02269
\(220\) 2.33014e8 1.47538
\(221\) −7.35190e7 −0.458170
\(222\) −1.49503e8 −0.917095
\(223\) −2.95302e8 −1.78320 −0.891599 0.452826i \(-0.850416\pi\)
−0.891599 + 0.452826i \(0.850416\pi\)
\(224\) −2.34977e8 −1.39688
\(225\) −9.41771e6 −0.0551196
\(226\) 2.08730e8 1.20283
\(227\) −4.08714e6 −0.0231915 −0.0115957 0.999933i \(-0.503691\pi\)
−0.0115957 + 0.999933i \(0.503691\pi\)
\(228\) −2.70773e8 −1.51298
\(229\) 2.00230e8 1.10181 0.550903 0.834570i \(-0.314284\pi\)
0.550903 + 0.834570i \(0.314284\pi\)
\(230\) 4.49044e7 0.243356
\(231\) −1.37378e8 −0.733290
\(232\) 2.26451e8 1.19060
\(233\) −9.85532e7 −0.510417 −0.255208 0.966886i \(-0.582144\pi\)
−0.255208 + 0.966886i \(0.582144\pi\)
\(234\) −7.46537e7 −0.380887
\(235\) −6.66787e7 −0.335158
\(236\) −3.95155e7 −0.195693
\(237\) 1.17240e8 0.572082
\(238\) −2.46775e8 −1.18654
\(239\) −3.06770e8 −1.45352 −0.726758 0.686894i \(-0.758974\pi\)
−0.726758 + 0.686894i \(0.758974\pi\)
\(240\) −2.75281e7 −0.128540
\(241\) 3.30373e8 1.52035 0.760176 0.649717i \(-0.225112\pi\)
0.760176 + 0.649717i \(0.225112\pi\)
\(242\) −5.38071e7 −0.244054
\(243\) 1.43489e7 0.0641500
\(244\) 4.49382e7 0.198039
\(245\) 8.36054e7 0.363206
\(246\) 2.03578e8 0.871881
\(247\) −2.98199e8 −1.25912
\(248\) −3.14188e8 −1.30800
\(249\) −1.11244e8 −0.456644
\(250\) 4.16143e8 1.68443
\(251\) −4.29400e8 −1.71397 −0.856987 0.515339i \(-0.827666\pi\)
−0.856987 + 0.515339i \(0.827666\pi\)
\(252\) −1.50476e8 −0.592333
\(253\) −4.65931e7 −0.180884
\(254\) 6.22158e8 2.38222
\(255\) −8.85998e7 −0.334612
\(256\) −1.54163e8 −0.574302
\(257\) −3.01963e8 −1.10966 −0.554828 0.831965i \(-0.687216\pi\)
−0.554828 + 0.831965i \(0.687216\pi\)
\(258\) 1.96359e8 0.711838
\(259\) −3.31869e8 −1.18691
\(260\) 2.81081e8 0.991803
\(261\) −1.43202e8 −0.498547
\(262\) 1.90224e8 0.653447
\(263\) −7.22951e7 −0.245055 −0.122528 0.992465i \(-0.539100\pi\)
−0.122528 + 0.992465i \(0.539100\pi\)
\(264\) −1.47619e8 −0.493775
\(265\) 1.88684e8 0.622836
\(266\) −1.00094e9 −3.26078
\(267\) −1.97826e8 −0.636055
\(268\) 6.81384e8 2.16232
\(269\) 1.46096e8 0.457622 0.228811 0.973471i \(-0.426516\pi\)
0.228811 + 0.973471i \(0.426516\pi\)
\(270\) −8.99672e7 −0.278171
\(271\) −2.87297e8 −0.876876 −0.438438 0.898761i \(-0.644468\pi\)
−0.438438 + 0.898761i \(0.644468\pi\)
\(272\) 5.13087e7 0.154597
\(273\) −1.65717e8 −0.492945
\(274\) 8.31362e8 2.44154
\(275\) −6.12693e7 −0.177656
\(276\) −5.10353e7 −0.146113
\(277\) 3.14837e8 0.890033 0.445016 0.895522i \(-0.353198\pi\)
0.445016 + 0.895522i \(0.353198\pi\)
\(278\) 3.14071e8 0.876741
\(279\) 1.98684e8 0.547708
\(280\) 3.15811e8 0.859753
\(281\) 7.28235e8 1.95794 0.978971 0.204002i \(-0.0653948\pi\)
0.978971 + 0.204002i \(0.0653948\pi\)
\(282\) 1.26198e8 0.335106
\(283\) −2.37367e8 −0.622541 −0.311271 0.950321i \(-0.600755\pi\)
−0.311271 + 0.950321i \(0.600755\pi\)
\(284\) 1.98244e8 0.513553
\(285\) −3.59368e8 −0.919564
\(286\) −4.85679e8 −1.22763
\(287\) 4.51904e8 1.12839
\(288\) 1.59671e8 0.393868
\(289\) −2.45200e8 −0.597556
\(290\) 8.97870e8 2.16182
\(291\) 3.95673e8 0.941263
\(292\) −1.13279e9 −2.66261
\(293\) −1.12840e7 −0.0262076 −0.0131038 0.999914i \(-0.504171\pi\)
−0.0131038 + 0.999914i \(0.504171\pi\)
\(294\) −1.58235e8 −0.363150
\(295\) −5.24446e7 −0.118939
\(296\) −3.56608e8 −0.799227
\(297\) 9.33505e7 0.206761
\(298\) −1.23629e9 −2.70622
\(299\) −5.62045e7 −0.121597
\(300\) −6.71109e7 −0.143505
\(301\) 4.35880e8 0.921264
\(302\) −1.77916e8 −0.371699
\(303\) 8.24388e7 0.170248
\(304\) 2.08112e8 0.424855
\(305\) 5.96416e7 0.120365
\(306\) 1.67687e8 0.334560
\(307\) 3.01581e8 0.594867 0.297433 0.954743i \(-0.403869\pi\)
0.297433 + 0.954743i \(0.403869\pi\)
\(308\) −9.78961e8 −1.90914
\(309\) −4.03069e8 −0.777186
\(310\) −1.24575e9 −2.37500
\(311\) 7.94458e8 1.49765 0.748824 0.662769i \(-0.230619\pi\)
0.748824 + 0.662769i \(0.230619\pi\)
\(312\) −1.78071e8 −0.331934
\(313\) 9.42174e8 1.73670 0.868352 0.495948i \(-0.165179\pi\)
0.868352 + 0.495948i \(0.165179\pi\)
\(314\) 3.10839e8 0.566607
\(315\) −1.99710e8 −0.360009
\(316\) 8.35459e8 1.48943
\(317\) −9.46323e7 −0.166852 −0.0834261 0.996514i \(-0.526586\pi\)
−0.0834261 + 0.996514i \(0.526586\pi\)
\(318\) −3.57110e8 −0.622740
\(319\) −9.31635e8 −1.60686
\(320\) −8.70626e8 −1.48527
\(321\) −5.99441e8 −1.01153
\(322\) −1.88657e8 −0.314903
\(323\) 6.69814e8 1.10598
\(324\) 1.02251e8 0.167016
\(325\) −7.39083e7 −0.119427
\(326\) 1.55082e9 2.47913
\(327\) −6.67719e7 −0.105603
\(328\) 4.85592e8 0.759824
\(329\) 2.80137e8 0.433696
\(330\) −5.85305e8 −0.896569
\(331\) 2.56756e8 0.389156 0.194578 0.980887i \(-0.437666\pi\)
0.194578 + 0.980887i \(0.437666\pi\)
\(332\) −7.92726e8 −1.18889
\(333\) 2.25510e8 0.334665
\(334\) 4.66202e8 0.684639
\(335\) 9.04327e8 1.31422
\(336\) 1.15654e8 0.166331
\(337\) 9.60763e7 0.136745 0.0683725 0.997660i \(-0.478219\pi\)
0.0683725 + 0.997660i \(0.478219\pi\)
\(338\) 5.37318e8 0.756873
\(339\) −3.14848e8 −0.438936
\(340\) −6.31365e8 −0.871172
\(341\) 1.29259e9 1.76531
\(342\) 6.80152e8 0.919422
\(343\) 5.32265e8 0.712195
\(344\) 4.68373e8 0.620350
\(345\) −6.77336e7 −0.0888050
\(346\) 5.12521e8 0.665189
\(347\) 4.17126e8 0.535937 0.267969 0.963428i \(-0.413648\pi\)
0.267969 + 0.963428i \(0.413648\pi\)
\(348\) −1.02046e9 −1.29798
\(349\) −5.98395e8 −0.753527 −0.376764 0.926309i \(-0.622963\pi\)
−0.376764 + 0.926309i \(0.622963\pi\)
\(350\) −2.48081e8 −0.309283
\(351\) 1.12607e8 0.138993
\(352\) 1.03878e9 1.26947
\(353\) −1.21639e9 −1.47184 −0.735921 0.677067i \(-0.763251\pi\)
−0.735921 + 0.677067i \(0.763251\pi\)
\(354\) 9.92586e7 0.118920
\(355\) 2.63107e8 0.312129
\(356\) −1.40972e9 −1.65599
\(357\) 3.72234e8 0.432989
\(358\) −1.61873e8 −0.186459
\(359\) −1.46927e8 −0.167599 −0.0837993 0.996483i \(-0.526705\pi\)
−0.0837993 + 0.996483i \(0.526705\pi\)
\(360\) −2.14598e8 −0.242419
\(361\) 1.82295e9 2.03939
\(362\) −1.61979e9 −1.79464
\(363\) 8.11624e7 0.0890598
\(364\) −1.18091e9 −1.28340
\(365\) −1.50342e9 −1.61829
\(366\) −1.12880e8 −0.120346
\(367\) −1.54258e9 −1.62899 −0.814494 0.580172i \(-0.802985\pi\)
−0.814494 + 0.580172i \(0.802985\pi\)
\(368\) 3.92250e7 0.0410295
\(369\) −3.07076e8 −0.318166
\(370\) −1.41394e9 −1.45119
\(371\) −7.92717e8 −0.805952
\(372\) 1.41583e9 1.42597
\(373\) 3.08393e8 0.307697 0.153849 0.988094i \(-0.450833\pi\)
0.153849 + 0.988094i \(0.450833\pi\)
\(374\) 1.09093e9 1.07832
\(375\) −6.27709e8 −0.614680
\(376\) 3.01020e8 0.292037
\(377\) −1.12382e9 −1.08019
\(378\) 3.77979e8 0.359954
\(379\) −8.49811e8 −0.801835 −0.400918 0.916114i \(-0.631309\pi\)
−0.400918 + 0.916114i \(0.631309\pi\)
\(380\) −2.56087e9 −2.39411
\(381\) −9.38461e8 −0.869318
\(382\) 1.34772e9 1.23703
\(383\) 8.83511e8 0.803556 0.401778 0.915737i \(-0.368392\pi\)
0.401778 + 0.915737i \(0.368392\pi\)
\(384\) 8.90822e8 0.802845
\(385\) −1.29927e9 −1.16034
\(386\) −5.43062e8 −0.480612
\(387\) −2.96187e8 −0.259763
\(388\) 2.81958e9 2.45061
\(389\) −1.07810e9 −0.928616 −0.464308 0.885674i \(-0.653697\pi\)
−0.464308 + 0.885674i \(0.653697\pi\)
\(390\) −7.06045e8 −0.602707
\(391\) 1.26247e8 0.106807
\(392\) −3.77435e8 −0.316476
\(393\) −2.86933e8 −0.238455
\(394\) −1.02986e9 −0.848285
\(395\) 1.10881e9 0.905250
\(396\) 6.65219e8 0.538309
\(397\) −2.25182e9 −1.80620 −0.903101 0.429427i \(-0.858715\pi\)
−0.903101 + 0.429427i \(0.858715\pi\)
\(398\) −5.91607e7 −0.0470373
\(399\) 1.50981e9 1.18992
\(400\) 5.15804e7 0.0402972
\(401\) 2.61153e8 0.202250 0.101125 0.994874i \(-0.467756\pi\)
0.101125 + 0.994874i \(0.467756\pi\)
\(402\) −1.71156e9 −1.31402
\(403\) 1.55924e9 1.18671
\(404\) 5.87461e8 0.443246
\(405\) 1.35706e8 0.101510
\(406\) −3.77222e9 −2.79741
\(407\) 1.46711e9 1.07866
\(408\) 3.99983e8 0.291562
\(409\) 3.01019e8 0.217552 0.108776 0.994066i \(-0.465307\pi\)
0.108776 + 0.994066i \(0.465307\pi\)
\(410\) 1.92536e9 1.37965
\(411\) −1.25402e9 −0.890963
\(412\) −2.87228e9 −2.02343
\(413\) 2.20335e8 0.153907
\(414\) 1.28195e8 0.0887913
\(415\) −1.05210e9 −0.722584
\(416\) 1.25306e9 0.853387
\(417\) −4.73744e8 −0.319939
\(418\) 4.42491e9 2.96338
\(419\) −2.40890e9 −1.59982 −0.799908 0.600123i \(-0.795118\pi\)
−0.799908 + 0.600123i \(0.795118\pi\)
\(420\) −1.42314e9 −0.937295
\(421\) 1.27032e9 0.829711 0.414855 0.909887i \(-0.363832\pi\)
0.414855 + 0.909887i \(0.363832\pi\)
\(422\) −1.33796e9 −0.866660
\(423\) −1.90357e8 −0.122286
\(424\) −8.51810e8 −0.542703
\(425\) 1.66013e8 0.104901
\(426\) −4.97966e8 −0.312080
\(427\) −2.50572e8 −0.155752
\(428\) −4.27164e9 −2.63355
\(429\) 7.32597e8 0.447986
\(430\) 1.85708e9 1.12640
\(431\) 2.24959e9 1.35342 0.676710 0.736250i \(-0.263405\pi\)
0.676710 + 0.736250i \(0.263405\pi\)
\(432\) −7.85884e7 −0.0468992
\(433\) 1.18429e9 0.701053 0.350527 0.936553i \(-0.386003\pi\)
0.350527 + 0.936553i \(0.386003\pi\)
\(434\) 5.23375e9 3.07326
\(435\) −1.35434e9 −0.788890
\(436\) −4.75819e8 −0.274940
\(437\) 5.12066e8 0.293522
\(438\) 2.84543e9 1.61804
\(439\) 1.33626e9 0.753818 0.376909 0.926250i \(-0.376987\pi\)
0.376909 + 0.926250i \(0.376987\pi\)
\(440\) −1.39612e9 −0.781339
\(441\) 2.38680e8 0.132520
\(442\) 1.31598e9 0.724886
\(443\) 1.47906e8 0.0808299 0.0404149 0.999183i \(-0.487132\pi\)
0.0404149 + 0.999183i \(0.487132\pi\)
\(444\) 1.60699e9 0.871310
\(445\) −1.87096e9 −1.00648
\(446\) 5.28585e9 2.82126
\(447\) 1.86482e9 0.987551
\(448\) 3.65776e9 1.92195
\(449\) 2.18342e9 1.13835 0.569174 0.822217i \(-0.307263\pi\)
0.569174 + 0.822217i \(0.307263\pi\)
\(450\) 1.68575e8 0.0872066
\(451\) −1.99776e9 −1.02548
\(452\) −2.24362e9 −1.14278
\(453\) 2.68368e8 0.135640
\(454\) 7.31589e7 0.0366920
\(455\) −1.56729e9 −0.780026
\(456\) 1.62236e9 0.801255
\(457\) −9.95575e8 −0.487941 −0.243971 0.969783i \(-0.578450\pi\)
−0.243971 + 0.969783i \(0.578450\pi\)
\(458\) −3.58408e9 −1.74320
\(459\) −2.52939e8 −0.122087
\(460\) −4.82672e8 −0.231206
\(461\) −3.16168e9 −1.50302 −0.751509 0.659722i \(-0.770674\pi\)
−0.751509 + 0.659722i \(0.770674\pi\)
\(462\) 2.45904e9 1.16016
\(463\) 3.01390e9 1.41122 0.705612 0.708599i \(-0.250672\pi\)
0.705612 + 0.708599i \(0.250672\pi\)
\(464\) 7.84310e8 0.364481
\(465\) 1.87908e9 0.866681
\(466\) 1.76408e9 0.807547
\(467\) 8.11607e8 0.368754 0.184377 0.982856i \(-0.440973\pi\)
0.184377 + 0.982856i \(0.440973\pi\)
\(468\) 8.02444e8 0.361871
\(469\) −3.79935e9 −1.70061
\(470\) 1.19354e9 0.530265
\(471\) −4.68869e8 −0.206765
\(472\) 2.36761e8 0.103636
\(473\) −1.92692e9 −0.837240
\(474\) −2.09858e9 −0.905110
\(475\) 6.73361e8 0.288284
\(476\) 2.65255e9 1.12730
\(477\) 5.38663e8 0.227249
\(478\) 5.49112e9 2.29966
\(479\) −2.44233e9 −1.01538 −0.507692 0.861538i \(-0.669501\pi\)
−0.507692 + 0.861538i \(0.669501\pi\)
\(480\) 1.51010e9 0.623249
\(481\) 1.76976e9 0.725113
\(482\) −5.91360e9 −2.40540
\(483\) 2.84569e8 0.114914
\(484\) 5.78366e8 0.231870
\(485\) 3.74212e9 1.48944
\(486\) −2.56843e8 −0.101494
\(487\) −1.11140e8 −0.0436032 −0.0218016 0.999762i \(-0.506940\pi\)
−0.0218016 + 0.999762i \(0.506940\pi\)
\(488\) −2.69251e8 −0.104879
\(489\) −2.33925e9 −0.904682
\(490\) −1.49652e9 −0.574640
\(491\) −1.10060e9 −0.419607 −0.209804 0.977744i \(-0.567282\pi\)
−0.209804 + 0.977744i \(0.567282\pi\)
\(492\) −2.18823e9 −0.828353
\(493\) 2.52432e9 0.948812
\(494\) 5.33770e9 1.99209
\(495\) 8.82872e8 0.327175
\(496\) −1.08819e9 −0.400422
\(497\) −1.10539e9 −0.403896
\(498\) 1.99124e9 0.722472
\(499\) 1.19831e9 0.431735 0.215867 0.976423i \(-0.430742\pi\)
0.215867 + 0.976423i \(0.430742\pi\)
\(500\) −4.47307e9 −1.60034
\(501\) −7.03218e8 −0.249838
\(502\) 7.68617e9 2.71173
\(503\) −4.77486e8 −0.167291 −0.0836456 0.996496i \(-0.526656\pi\)
−0.0836456 + 0.996496i \(0.526656\pi\)
\(504\) 9.01590e8 0.313691
\(505\) 7.79673e8 0.269397
\(506\) 8.34006e8 0.286182
\(507\) −8.10489e8 −0.276197
\(508\) −6.68750e9 −2.26329
\(509\) −1.61594e9 −0.543140 −0.271570 0.962419i \(-0.587543\pi\)
−0.271570 + 0.962419i \(0.587543\pi\)
\(510\) 1.58592e9 0.529401
\(511\) 6.31633e9 2.09407
\(512\) −1.46367e9 −0.481945
\(513\) −1.02594e9 −0.335514
\(514\) 5.40509e9 1.75562
\(515\) −3.81207e9 −1.22980
\(516\) −2.11064e9 −0.676300
\(517\) −1.23842e9 −0.394140
\(518\) 5.94038e9 1.87785
\(519\) −7.73085e8 −0.242740
\(520\) −1.68412e9 −0.525245
\(521\) 2.28015e9 0.706368 0.353184 0.935554i \(-0.385099\pi\)
0.353184 + 0.935554i \(0.385099\pi\)
\(522\) 2.56328e9 0.788768
\(523\) −4.17653e8 −0.127661 −0.0638307 0.997961i \(-0.520332\pi\)
−0.0638307 + 0.997961i \(0.520332\pi\)
\(524\) −2.04469e9 −0.620824
\(525\) 3.74205e8 0.112863
\(526\) 1.29407e9 0.387710
\(527\) −3.50235e9 −1.04237
\(528\) −5.11277e8 −0.151160
\(529\) −3.30831e9 −0.971654
\(530\) −3.37740e9 −0.985410
\(531\) −1.49721e8 −0.0433963
\(532\) 1.07590e10 3.09799
\(533\) −2.40987e9 −0.689364
\(534\) 3.54105e9 1.00632
\(535\) −5.66928e9 −1.60062
\(536\) −4.08257e9 −1.14514
\(537\) 2.44169e8 0.0680425
\(538\) −2.61510e9 −0.724019
\(539\) 1.55280e9 0.427124
\(540\) 9.67047e8 0.264283
\(541\) 7.14439e9 1.93988 0.969940 0.243346i \(-0.0782450\pi\)
0.969940 + 0.243346i \(0.0782450\pi\)
\(542\) 5.14255e9 1.38734
\(543\) 2.44328e9 0.654898
\(544\) −2.81463e9 −0.749592
\(545\) −6.31502e8 −0.167104
\(546\) 2.96631e9 0.779905
\(547\) −1.90559e9 −0.497822 −0.248911 0.968526i \(-0.580073\pi\)
−0.248911 + 0.968526i \(0.580073\pi\)
\(548\) −8.93622e9 −2.31965
\(549\) 1.70267e8 0.0439166
\(550\) 1.09671e9 0.281075
\(551\) 1.02388e10 2.60748
\(552\) 3.05783e8 0.0773795
\(553\) −4.65846e9 −1.17140
\(554\) −5.63551e9 −1.40815
\(555\) 2.13278e9 0.529567
\(556\) −3.37592e9 −0.832971
\(557\) −8.13950e9 −1.99574 −0.997872 0.0652056i \(-0.979230\pi\)
−0.997872 + 0.0652056i \(0.979230\pi\)
\(558\) −3.55641e9 −0.866547
\(559\) −2.32442e9 −0.562824
\(560\) 1.09381e9 0.263198
\(561\) −1.64556e9 −0.393499
\(562\) −1.30353e10 −3.09772
\(563\) 3.38360e9 0.799098 0.399549 0.916712i \(-0.369167\pi\)
0.399549 + 0.916712i \(0.369167\pi\)
\(564\) −1.35649e9 −0.318376
\(565\) −2.97771e9 −0.694564
\(566\) 4.24883e9 0.984944
\(567\) −5.70143e8 −0.131354
\(568\) −1.18779e9 −0.271971
\(569\) 1.41168e9 0.321250 0.160625 0.987016i \(-0.448649\pi\)
0.160625 + 0.987016i \(0.448649\pi\)
\(570\) 6.43261e9 1.45487
\(571\) 3.06389e9 0.688726 0.344363 0.938837i \(-0.388095\pi\)
0.344363 + 0.938837i \(0.388095\pi\)
\(572\) 5.22051e9 1.16634
\(573\) −2.03290e9 −0.451414
\(574\) −8.08900e9 −1.78527
\(575\) 1.26915e8 0.0278404
\(576\) −2.48550e9 −0.541921
\(577\) 5.11879e9 1.10931 0.554653 0.832082i \(-0.312851\pi\)
0.554653 + 0.832082i \(0.312851\pi\)
\(578\) 4.38904e9 0.945414
\(579\) 8.19154e8 0.175384
\(580\) −9.65110e9 −2.05390
\(581\) 4.42018e9 0.935026
\(582\) −7.08247e9 −1.48920
\(583\) 3.50441e9 0.732445
\(584\) 6.78718e9 1.41008
\(585\) 1.06500e9 0.219939
\(586\) 2.01982e8 0.0414639
\(587\) 5.18384e9 1.05784 0.528918 0.848673i \(-0.322598\pi\)
0.528918 + 0.848673i \(0.322598\pi\)
\(588\) 1.70084e9 0.345020
\(589\) −1.42058e10 −2.86459
\(590\) 9.38748e8 0.188177
\(591\) 1.55344e9 0.309555
\(592\) −1.23511e9 −0.244669
\(593\) 5.88946e9 1.15980 0.579902 0.814686i \(-0.303091\pi\)
0.579902 + 0.814686i \(0.303091\pi\)
\(594\) −1.67096e9 −0.327124
\(595\) 3.52044e9 0.685153
\(596\) 1.32887e10 2.57112
\(597\) 8.92377e7 0.0171648
\(598\) 1.00605e9 0.192382
\(599\) 1.29167e9 0.245561 0.122780 0.992434i \(-0.460819\pi\)
0.122780 + 0.992434i \(0.460819\pi\)
\(600\) 4.02101e8 0.0759986
\(601\) −4.79647e9 −0.901282 −0.450641 0.892705i \(-0.648805\pi\)
−0.450641 + 0.892705i \(0.648805\pi\)
\(602\) −7.80216e9 −1.45756
\(603\) 2.58171e9 0.479510
\(604\) 1.91240e9 0.353142
\(605\) 7.67602e8 0.140926
\(606\) −1.47564e9 −0.269355
\(607\) −3.30008e9 −0.598914 −0.299457 0.954110i \(-0.596805\pi\)
−0.299457 + 0.954110i \(0.596805\pi\)
\(608\) −1.14164e10 −2.05999
\(609\) 5.69000e9 1.02083
\(610\) −1.06757e9 −0.190433
\(611\) −1.49389e9 −0.264956
\(612\) −1.80245e9 −0.317858
\(613\) 2.21730e9 0.388788 0.194394 0.980924i \(-0.437726\pi\)
0.194394 + 0.980924i \(0.437726\pi\)
\(614\) −5.39824e9 −0.941159
\(615\) −2.90420e9 −0.503459
\(616\) 5.86553e9 1.01105
\(617\) −3.08716e9 −0.529128 −0.264564 0.964368i \(-0.585228\pi\)
−0.264564 + 0.964368i \(0.585228\pi\)
\(618\) 7.21486e9 1.22961
\(619\) 4.78027e9 0.810094 0.405047 0.914296i \(-0.367255\pi\)
0.405047 + 0.914296i \(0.367255\pi\)
\(620\) 1.33904e10 2.25643
\(621\) −1.93369e8 −0.0324016
\(622\) −1.42206e10 −2.36948
\(623\) 7.86047e9 1.30239
\(624\) −6.16747e8 −0.101616
\(625\) −4.92735e9 −0.807298
\(626\) −1.68647e10 −2.74770
\(627\) −6.67452e9 −1.08139
\(628\) −3.34117e9 −0.538319
\(629\) −3.97522e9 −0.636919
\(630\) 3.57478e9 0.569583
\(631\) −4.87262e9 −0.772075 −0.386037 0.922483i \(-0.626156\pi\)
−0.386037 + 0.922483i \(0.626156\pi\)
\(632\) −5.00573e9 −0.788782
\(633\) 2.01817e9 0.316260
\(634\) 1.69390e9 0.263982
\(635\) −8.87559e9 −1.37559
\(636\) 3.83853e9 0.591650
\(637\) 1.87312e9 0.287129
\(638\) 1.66761e10 2.54227
\(639\) 7.51131e8 0.113884
\(640\) 8.42504e9 1.27040
\(641\) 2.85862e9 0.428700 0.214350 0.976757i \(-0.431237\pi\)
0.214350 + 0.976757i \(0.431237\pi\)
\(642\) 1.07299e10 1.60038
\(643\) −7.22075e9 −1.07113 −0.535567 0.844493i \(-0.679902\pi\)
−0.535567 + 0.844493i \(0.679902\pi\)
\(644\) 2.02785e9 0.299182
\(645\) −2.80122e9 −0.411044
\(646\) −1.19895e10 −1.74980
\(647\) −4.74606e8 −0.0688919 −0.0344460 0.999407i \(-0.510967\pi\)
−0.0344460 + 0.999407i \(0.510967\pi\)
\(648\) −6.12644e8 −0.0884496
\(649\) −9.74051e8 −0.139870
\(650\) 1.32294e9 0.188949
\(651\) −7.89457e9 −1.12149
\(652\) −1.66696e10 −2.35536
\(653\) −2.58487e9 −0.363281 −0.181640 0.983365i \(-0.558141\pi\)
−0.181640 + 0.983365i \(0.558141\pi\)
\(654\) 1.19520e9 0.167078
\(655\) −2.71370e9 −0.377326
\(656\) 1.68184e9 0.232607
\(657\) −4.29204e9 −0.590453
\(658\) −5.01440e9 −0.686164
\(659\) 2.68455e9 0.365404 0.182702 0.983168i \(-0.441516\pi\)
0.182702 + 0.983168i \(0.441516\pi\)
\(660\) 6.29138e9 0.851808
\(661\) −9.19497e9 −1.23835 −0.619177 0.785251i \(-0.712534\pi\)
−0.619177 + 0.785251i \(0.712534\pi\)
\(662\) −4.59589e9 −0.615696
\(663\) −1.98501e9 −0.264525
\(664\) 4.74969e9 0.629618
\(665\) 1.42792e10 1.88290
\(666\) −4.03658e9 −0.529485
\(667\) 1.92982e9 0.251812
\(668\) −5.01115e9 −0.650459
\(669\) −7.97316e9 −1.02953
\(670\) −1.61873e10 −2.07928
\(671\) 1.10772e9 0.141547
\(672\) −6.34438e9 −0.806486
\(673\) 1.23253e10 1.55863 0.779317 0.626630i \(-0.215566\pi\)
0.779317 + 0.626630i \(0.215566\pi\)
\(674\) −1.71975e9 −0.216349
\(675\) −2.54278e8 −0.0318233
\(676\) −5.77557e9 −0.719087
\(677\) 1.51222e10 1.87307 0.936537 0.350569i \(-0.114012\pi\)
0.936537 + 0.350569i \(0.114012\pi\)
\(678\) 5.63571e9 0.694456
\(679\) −1.57218e10 −1.92733
\(680\) 3.78288e9 0.461361
\(681\) −1.10353e8 −0.0133896
\(682\) −2.31371e10 −2.79296
\(683\) −2.68739e9 −0.322745 −0.161372 0.986894i \(-0.551592\pi\)
−0.161372 + 0.986894i \(0.551592\pi\)
\(684\) −7.31088e9 −0.873521
\(685\) −1.18601e10 −1.40984
\(686\) −9.52744e9 −1.12679
\(687\) 5.40621e9 0.636127
\(688\) 1.62220e9 0.189909
\(689\) 4.22732e9 0.492377
\(690\) 1.21242e9 0.140501
\(691\) −9.07091e9 −1.04587 −0.522935 0.852373i \(-0.675163\pi\)
−0.522935 + 0.852373i \(0.675163\pi\)
\(692\) −5.50903e9 −0.631980
\(693\) −3.70921e9 −0.423365
\(694\) −7.46647e9 −0.847924
\(695\) −4.48048e9 −0.506265
\(696\) 6.11417e9 0.687393
\(697\) 5.41305e9 0.605518
\(698\) 1.07111e10 1.19218
\(699\) −2.66094e9 −0.294689
\(700\) 2.66660e9 0.293842
\(701\) 2.72996e9 0.299326 0.149663 0.988737i \(-0.452181\pi\)
0.149663 + 0.988737i \(0.452181\pi\)
\(702\) −2.01565e9 −0.219905
\(703\) −1.61238e10 −1.75035
\(704\) −1.61701e10 −1.74666
\(705\) −1.80032e9 −0.193504
\(706\) 2.17731e10 2.32865
\(707\) −3.27564e9 −0.348601
\(708\) −1.06692e9 −0.112983
\(709\) −9.03726e8 −0.0952302 −0.0476151 0.998866i \(-0.515162\pi\)
−0.0476151 + 0.998866i \(0.515162\pi\)
\(710\) −4.70957e9 −0.493829
\(711\) 3.16549e9 0.330292
\(712\) 8.44644e9 0.876988
\(713\) −2.67752e9 −0.276642
\(714\) −6.66292e9 −0.685047
\(715\) 6.92861e9 0.708883
\(716\) 1.73996e9 0.177151
\(717\) −8.28278e9 −0.839188
\(718\) 2.62996e9 0.265164
\(719\) −6.51161e9 −0.653337 −0.326668 0.945139i \(-0.605926\pi\)
−0.326668 + 0.945139i \(0.605926\pi\)
\(720\) −7.43258e8 −0.0742123
\(721\) 1.60156e10 1.59137
\(722\) −3.26304e10 −3.22658
\(723\) 8.92006e9 0.877776
\(724\) 1.74109e10 1.70505
\(725\) 2.53769e9 0.247318
\(726\) −1.45279e9 −0.140905
\(727\) −6.14813e9 −0.593435 −0.296717 0.954965i \(-0.595892\pi\)
−0.296717 + 0.954965i \(0.595892\pi\)
\(728\) 7.07550e9 0.679669
\(729\) 3.87420e8 0.0370370
\(730\) 2.69110e10 2.56035
\(731\) 5.22110e9 0.494369
\(732\) 1.21333e9 0.114338
\(733\) 2.71313e9 0.254452 0.127226 0.991874i \(-0.459393\pi\)
0.127226 + 0.991874i \(0.459393\pi\)
\(734\) 2.76120e10 2.57728
\(735\) 2.25735e9 0.209697
\(736\) −2.15176e9 −0.198939
\(737\) 1.67960e10 1.54550
\(738\) 5.49660e9 0.503381
\(739\) 1.10553e10 1.00766 0.503831 0.863802i \(-0.331924\pi\)
0.503831 + 0.863802i \(0.331924\pi\)
\(740\) 1.51983e10 1.37874
\(741\) −8.05137e9 −0.726953
\(742\) 1.41895e10 1.27512
\(743\) 1.67697e10 1.49991 0.749954 0.661490i \(-0.230075\pi\)
0.749954 + 0.661490i \(0.230075\pi\)
\(744\) −8.48307e9 −0.755176
\(745\) 1.76367e10 1.56268
\(746\) −5.52017e9 −0.486818
\(747\) −3.00358e9 −0.263644
\(748\) −1.17263e10 −1.02448
\(749\) 2.38183e10 2.07121
\(750\) 1.12359e10 0.972506
\(751\) 7.82494e9 0.674126 0.337063 0.941482i \(-0.390566\pi\)
0.337063 + 0.941482i \(0.390566\pi\)
\(752\) 1.04258e9 0.0894020
\(753\) −1.15938e10 −0.989563
\(754\) 2.01161e10 1.70901
\(755\) 2.53812e9 0.214634
\(756\) −4.06285e9 −0.341983
\(757\) 1.40665e10 1.17856 0.589279 0.807930i \(-0.299412\pi\)
0.589279 + 0.807930i \(0.299412\pi\)
\(758\) 1.52115e10 1.26861
\(759\) −1.25801e9 −0.104433
\(760\) 1.53437e10 1.26789
\(761\) −4.85071e9 −0.398987 −0.199493 0.979899i \(-0.563930\pi\)
−0.199493 + 0.979899i \(0.563930\pi\)
\(762\) 1.67983e10 1.37538
\(763\) 2.65313e9 0.216233
\(764\) −1.44865e10 −1.17527
\(765\) −2.39219e9 −0.193189
\(766\) −1.58147e10 −1.27133
\(767\) −1.17498e9 −0.0940260
\(768\) −4.16240e9 −0.331574
\(769\) 9.60096e9 0.761329 0.380665 0.924713i \(-0.375695\pi\)
0.380665 + 0.924713i \(0.375695\pi\)
\(770\) 2.32566e10 1.83582
\(771\) −8.15301e9 −0.640660
\(772\) 5.83731e9 0.456617
\(773\) −2.04008e10 −1.58862 −0.794309 0.607514i \(-0.792167\pi\)
−0.794309 + 0.607514i \(0.792167\pi\)
\(774\) 5.30169e9 0.410980
\(775\) −3.52090e9 −0.271705
\(776\) −1.68938e10 −1.29781
\(777\) −8.96045e9 −0.685262
\(778\) 1.92978e10 1.46919
\(779\) 2.19558e10 1.66405
\(780\) 7.58920e9 0.572618
\(781\) 4.88668e9 0.367058
\(782\) −2.25979e9 −0.168983
\(783\) −3.86644e9 −0.287836
\(784\) −1.30724e9 −0.0968836
\(785\) −4.43437e9 −0.327181
\(786\) 5.13604e9 0.377268
\(787\) 1.60432e10 1.17322 0.586608 0.809871i \(-0.300463\pi\)
0.586608 + 0.809871i \(0.300463\pi\)
\(788\) 1.10699e10 0.805935
\(789\) −1.95197e9 −0.141483
\(790\) −1.98475e10 −1.43223
\(791\) 1.25102e10 0.898768
\(792\) −3.98572e9 −0.285081
\(793\) 1.33623e9 0.0951532
\(794\) 4.03071e10 2.85765
\(795\) 5.09446e9 0.359595
\(796\) 6.35911e8 0.0446890
\(797\) −1.85146e10 −1.29542 −0.647711 0.761886i \(-0.724273\pi\)
−0.647711 + 0.761886i \(0.724273\pi\)
\(798\) −2.70253e10 −1.88261
\(799\) 3.35557e9 0.232730
\(800\) −2.82953e9 −0.195389
\(801\) −5.34131e9 −0.367226
\(802\) −4.67458e9 −0.319987
\(803\) −2.79230e10 −1.90308
\(804\) 1.83974e10 1.24842
\(805\) 2.69134e9 0.181838
\(806\) −2.79100e10 −1.87753
\(807\) 3.94460e9 0.264208
\(808\) −3.51983e9 −0.234737
\(809\) 1.01950e9 0.0676968 0.0338484 0.999427i \(-0.489224\pi\)
0.0338484 + 0.999427i \(0.489224\pi\)
\(810\) −2.42912e9 −0.160602
\(811\) 2.10382e10 1.38496 0.692478 0.721439i \(-0.256519\pi\)
0.692478 + 0.721439i \(0.256519\pi\)
\(812\) 4.05472e10 2.65775
\(813\) −7.75701e9 −0.506265
\(814\) −2.62610e10 −1.70658
\(815\) −2.21237e10 −1.43155
\(816\) 1.38534e9 0.0892564
\(817\) 2.11772e10 1.35860
\(818\) −5.38818e9 −0.344196
\(819\) −4.47437e9 −0.284602
\(820\) −2.06954e10 −1.31077
\(821\) 4.64433e9 0.292902 0.146451 0.989218i \(-0.453215\pi\)
0.146451 + 0.989218i \(0.453215\pi\)
\(822\) 2.24468e10 1.40962
\(823\) −5.01145e9 −0.313375 −0.156687 0.987648i \(-0.550081\pi\)
−0.156687 + 0.987648i \(0.550081\pi\)
\(824\) 1.72095e10 1.07158
\(825\) −1.65427e9 −0.102569
\(826\) −3.94396e9 −0.243502
\(827\) −2.72122e10 −1.67300 −0.836498 0.547970i \(-0.815401\pi\)
−0.836498 + 0.547970i \(0.815401\pi\)
\(828\) −1.37795e9 −0.0843585
\(829\) 3.17275e10 1.93417 0.967086 0.254450i \(-0.0818943\pi\)
0.967086 + 0.254450i \(0.0818943\pi\)
\(830\) 1.88324e10 1.14322
\(831\) 8.50059e9 0.513861
\(832\) −1.95057e10 −1.17417
\(833\) −4.20739e9 −0.252206
\(834\) 8.47993e9 0.506187
\(835\) −6.65076e9 −0.395338
\(836\) −4.75628e10 −2.81544
\(837\) 5.36448e9 0.316219
\(838\) 4.31189e10 2.53112
\(839\) −3.12839e9 −0.182875 −0.0914374 0.995811i \(-0.529146\pi\)
−0.0914374 + 0.995811i \(0.529146\pi\)
\(840\) 8.52689e9 0.496378
\(841\) 2.13371e10 1.23694
\(842\) −2.27385e10 −1.31271
\(843\) 1.96624e10 1.13042
\(844\) 1.43815e10 0.823393
\(845\) −7.66529e9 −0.437049
\(846\) 3.40736e9 0.193474
\(847\) −3.22492e9 −0.182359
\(848\) −2.95024e9 −0.166139
\(849\) −6.40892e9 −0.359424
\(850\) −2.97160e9 −0.165968
\(851\) −3.03902e9 −0.169036
\(852\) 5.35258e9 0.296500
\(853\) −3.10920e10 −1.71525 −0.857623 0.514278i \(-0.828060\pi\)
−0.857623 + 0.514278i \(0.828060\pi\)
\(854\) 4.48519e9 0.246421
\(855\) −9.70293e9 −0.530911
\(856\) 2.55939e10 1.39469
\(857\) −6.63199e9 −0.359924 −0.179962 0.983674i \(-0.557597\pi\)
−0.179962 + 0.983674i \(0.557597\pi\)
\(858\) −1.31133e10 −0.708774
\(859\) 2.04251e9 0.109948 0.0549741 0.998488i \(-0.482492\pi\)
0.0549741 + 0.998488i \(0.482492\pi\)
\(860\) −1.99616e10 −1.07016
\(861\) 1.22014e10 0.651477
\(862\) −4.02672e10 −2.14129
\(863\) 6.09687e8 0.0322901 0.0161450 0.999870i \(-0.494861\pi\)
0.0161450 + 0.999870i \(0.494861\pi\)
\(864\) 4.31110e9 0.227400
\(865\) −7.31153e9 −0.384107
\(866\) −2.11986e10 −1.10916
\(867\) −6.62041e9 −0.344999
\(868\) −5.62569e10 −2.91983
\(869\) 2.05939e10 1.06456
\(870\) 2.42425e10 1.24813
\(871\) 2.02608e10 1.03895
\(872\) 2.85091e9 0.145605
\(873\) 1.06832e10 0.543439
\(874\) −9.16589e9 −0.464392
\(875\) 2.49415e10 1.25862
\(876\) −3.05852e10 −1.53726
\(877\) −1.17372e10 −0.587580 −0.293790 0.955870i \(-0.594917\pi\)
−0.293790 + 0.955870i \(0.594917\pi\)
\(878\) −2.39189e10 −1.19264
\(879\) −3.04668e8 −0.0151310
\(880\) −4.83546e9 −0.239193
\(881\) 3.53359e10 1.74101 0.870504 0.492162i \(-0.163793\pi\)
0.870504 + 0.492162i \(0.163793\pi\)
\(882\) −4.27233e9 −0.209665
\(883\) 2.20915e10 1.07985 0.539924 0.841714i \(-0.318453\pi\)
0.539924 + 0.841714i \(0.318453\pi\)
\(884\) −1.41453e10 −0.688697
\(885\) −1.41600e9 −0.0686694
\(886\) −2.64748e9 −0.127884
\(887\) −2.64826e10 −1.27417 −0.637085 0.770793i \(-0.719860\pi\)
−0.637085 + 0.770793i \(0.719860\pi\)
\(888\) −9.62842e9 −0.461434
\(889\) 3.72890e10 1.78002
\(890\) 3.34899e10 1.59239
\(891\) 2.52046e9 0.119374
\(892\) −5.68170e10 −2.68041
\(893\) 1.36105e10 0.639577
\(894\) −3.33798e10 −1.56244
\(895\) 2.30925e9 0.107669
\(896\) −3.53961e10 −1.64391
\(897\) −1.51752e9 −0.0702040
\(898\) −3.90828e10 −1.80102
\(899\) −5.35373e10 −2.45752
\(900\) −1.81199e9 −0.0828529
\(901\) −9.49541e9 −0.432491
\(902\) 3.57595e10 1.62244
\(903\) 1.17688e10 0.531892
\(904\) 1.34428e10 0.605202
\(905\) 2.31076e10 1.03630
\(906\) −4.80374e9 −0.214600
\(907\) −1.70454e10 −0.758547 −0.379274 0.925285i \(-0.623826\pi\)
−0.379274 + 0.925285i \(0.623826\pi\)
\(908\) −7.86377e8 −0.0348602
\(909\) 2.22585e9 0.0982928
\(910\) 2.80542e10 1.23411
\(911\) −3.21064e10 −1.40695 −0.703473 0.710722i \(-0.748368\pi\)
−0.703473 + 0.710722i \(0.748368\pi\)
\(912\) 5.61903e9 0.245290
\(913\) −1.95406e10 −0.849747
\(914\) 1.78206e10 0.771988
\(915\) 1.61032e9 0.0694927
\(916\) 3.85248e10 1.65618
\(917\) 1.14011e10 0.488261
\(918\) 4.52755e9 0.193159
\(919\) 9.83116e8 0.0417831 0.0208915 0.999782i \(-0.493350\pi\)
0.0208915 + 0.999782i \(0.493350\pi\)
\(920\) 2.89197e9 0.122444
\(921\) 8.14269e9 0.343447
\(922\) 5.65934e10 2.37798
\(923\) 5.89473e9 0.246750
\(924\) −2.64319e10 −1.10224
\(925\) −3.99628e9 −0.166020
\(926\) −5.39483e10 −2.23274
\(927\) −1.08829e10 −0.448709
\(928\) −4.30247e10 −1.76726
\(929\) 1.58139e10 0.647119 0.323559 0.946208i \(-0.395120\pi\)
0.323559 + 0.946208i \(0.395120\pi\)
\(930\) −3.36351e10 −1.37121
\(931\) −1.70655e10 −0.693100
\(932\) −1.89619e10 −0.767231
\(933\) 2.14504e10 0.864668
\(934\) −1.45276e10 −0.583418
\(935\) −1.55630e10 −0.622664
\(936\) −4.80791e9 −0.191642
\(937\) 1.93290e10 0.767576 0.383788 0.923421i \(-0.374619\pi\)
0.383788 + 0.923421i \(0.374619\pi\)
\(938\) 6.80076e10 2.69059
\(939\) 2.54387e10 1.00269
\(940\) −1.28292e10 −0.503792
\(941\) −1.84380e9 −0.0721356 −0.0360678 0.999349i \(-0.511483\pi\)
−0.0360678 + 0.999349i \(0.511483\pi\)
\(942\) 8.39265e9 0.327131
\(943\) 4.13822e9 0.160703
\(944\) 8.20018e8 0.0317264
\(945\) −5.39218e9 −0.207852
\(946\) 3.44915e10 1.32463
\(947\) −3.45923e10 −1.32359 −0.661797 0.749683i \(-0.730206\pi\)
−0.661797 + 0.749683i \(0.730206\pi\)
\(948\) 2.25574e10 0.859923
\(949\) −3.36831e10 −1.27932
\(950\) −1.20530e10 −0.456104
\(951\) −2.55507e9 −0.0963321
\(952\) −1.58930e10 −0.597003
\(953\) −2.99406e10 −1.12056 −0.560281 0.828303i \(-0.689307\pi\)
−0.560281 + 0.828303i \(0.689307\pi\)
\(954\) −9.64196e9 −0.359539
\(955\) −1.92264e10 −0.714309
\(956\) −5.90234e10 −2.18485
\(957\) −2.51542e10 −0.927722
\(958\) 4.37173e10 1.60647
\(959\) 4.98277e10 1.82434
\(960\) −2.35069e10 −0.857524
\(961\) 4.67675e10 1.69986
\(962\) −3.16783e10 −1.14723
\(963\) −1.61849e10 −0.584007
\(964\) 6.35646e10 2.28531
\(965\) 7.74723e9 0.277524
\(966\) −5.09373e9 −0.181809
\(967\) 3.35194e9 0.119207 0.0596037 0.998222i \(-0.481016\pi\)
0.0596037 + 0.998222i \(0.481016\pi\)
\(968\) −3.46533e9 −0.122795
\(969\) 1.80850e10 0.638535
\(970\) −6.69832e10 −2.35649
\(971\) 3.76669e10 1.32036 0.660180 0.751107i \(-0.270480\pi\)
0.660180 + 0.751107i \(0.270480\pi\)
\(972\) 2.76077e9 0.0964269
\(973\) 1.88239e10 0.655109
\(974\) 1.98938e9 0.0689861
\(975\) −1.99552e9 −0.0689510
\(976\) −9.32548e8 −0.0321068
\(977\) 2.34371e10 0.804030 0.402015 0.915633i \(-0.368310\pi\)
0.402015 + 0.915633i \(0.368310\pi\)
\(978\) 4.18721e10 1.43133
\(979\) −3.47493e10 −1.18360
\(980\) 1.60859e10 0.545952
\(981\) −1.80284e9 −0.0609699
\(982\) 1.97005e10 0.663875
\(983\) 3.82382e10 1.28399 0.641993 0.766710i \(-0.278108\pi\)
0.641993 + 0.766710i \(0.278108\pi\)
\(984\) 1.31110e10 0.438685
\(985\) 1.46918e10 0.489833
\(986\) −4.51848e10 −1.50115
\(987\) 7.56370e9 0.250394
\(988\) −5.73743e10 −1.89264
\(989\) 3.99148e9 0.131204
\(990\) −1.58032e10 −0.517634
\(991\) 5.46278e10 1.78302 0.891510 0.453001i \(-0.149647\pi\)
0.891510 + 0.453001i \(0.149647\pi\)
\(992\) 5.96944e10 1.94152
\(993\) 6.93243e9 0.224679
\(994\) 1.97863e10 0.639017
\(995\) 8.43975e8 0.0271612
\(996\) −2.14036e10 −0.686403
\(997\) −2.14509e10 −0.685507 −0.342753 0.939425i \(-0.611360\pi\)
−0.342753 + 0.939425i \(0.611360\pi\)
\(998\) −2.14495e10 −0.683062
\(999\) 6.08876e9 0.193219
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.1.4 17
3.2 odd 2 531.8.a.d.1.14 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.4 17 1.1 even 1 trivial
531.8.a.d.1.14 17 3.2 odd 2