Properties

Label 177.8.a.c.1.13
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(11.5967\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+11.5967 q^{2} -27.0000 q^{3} +6.48306 q^{4} -401.104 q^{5} -313.110 q^{6} +213.573 q^{7} -1409.19 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+11.5967 q^{2} -27.0000 q^{3} +6.48306 q^{4} -401.104 q^{5} -313.110 q^{6} +213.573 q^{7} -1409.19 q^{8} +729.000 q^{9} -4651.48 q^{10} -4974.14 q^{11} -175.043 q^{12} -12649.9 q^{13} +2476.74 q^{14} +10829.8 q^{15} -17171.8 q^{16} +36937.0 q^{17} +8453.98 q^{18} -31065.7 q^{19} -2600.38 q^{20} -5766.47 q^{21} -57683.5 q^{22} +52594.0 q^{23} +38048.2 q^{24} +82759.6 q^{25} -146697. q^{26} -19683.0 q^{27} +1384.61 q^{28} -209912. q^{29} +125590. q^{30} -11865.2 q^{31} -18759.2 q^{32} +134302. q^{33} +428347. q^{34} -85665.0 q^{35} +4726.15 q^{36} +16919.7 q^{37} -360259. q^{38} +341549. q^{39} +565233. q^{40} +801040. q^{41} -66871.9 q^{42} +720238. q^{43} -32247.7 q^{44} -292405. q^{45} +609916. q^{46} -54403.0 q^{47} +463639. q^{48} -777930. q^{49} +959737. q^{50} -997299. q^{51} -82010.4 q^{52} -295498. q^{53} -228258. q^{54} +1.99515e6 q^{55} -300965. q^{56} +838773. q^{57} -2.43428e6 q^{58} -205379. q^{59} +70210.3 q^{60} +1.37027e6 q^{61} -137597. q^{62} +155695. q^{63} +1.98045e6 q^{64} +5.07395e6 q^{65} +1.55746e6 q^{66} +3.05207e6 q^{67} +239465. q^{68} -1.42004e6 q^{69} -993430. q^{70} -4.70310e6 q^{71} -1.02730e6 q^{72} -3.65576e6 q^{73} +196212. q^{74} -2.23451e6 q^{75} -201401. q^{76} -1.06234e6 q^{77} +3.96083e6 q^{78} +6.37313e6 q^{79} +6.88768e6 q^{80} +531441. q^{81} +9.28941e6 q^{82} -2.08701e6 q^{83} -37384.4 q^{84} -1.48156e7 q^{85} +8.35238e6 q^{86} +5.66762e6 q^{87} +7.00953e6 q^{88} +1.43644e6 q^{89} -3.39093e6 q^{90} -2.70169e6 q^{91} +340970. q^{92} +320360. q^{93} -630894. q^{94} +1.24606e7 q^{95} +506498. q^{96} -6.42693e6 q^{97} -9.02140e6 q^{98} -3.62615e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.5967 1.02501 0.512506 0.858684i \(-0.328717\pi\)
0.512506 + 0.858684i \(0.328717\pi\)
\(3\) −27.0000 −0.577350
\(4\) 6.48306 0.0506489
\(5\) −401.104 −1.43503 −0.717517 0.696541i \(-0.754721\pi\)
−0.717517 + 0.696541i \(0.754721\pi\)
\(6\) −313.110 −0.591791
\(7\) 213.573 0.235344 0.117672 0.993053i \(-0.462457\pi\)
0.117672 + 0.993053i \(0.462457\pi\)
\(8\) −1409.19 −0.973096
\(9\) 729.000 0.333333
\(10\) −4651.48 −1.47093
\(11\) −4974.14 −1.12679 −0.563396 0.826187i \(-0.690505\pi\)
−0.563396 + 0.826187i \(0.690505\pi\)
\(12\) −175.043 −0.0292422
\(13\) −12649.9 −1.59693 −0.798467 0.602038i \(-0.794355\pi\)
−0.798467 + 0.602038i \(0.794355\pi\)
\(14\) 2476.74 0.241230
\(15\) 10829.8 0.828517
\(16\) −17171.8 −1.04808
\(17\) 36937.0 1.82344 0.911718 0.410816i \(-0.134756\pi\)
0.911718 + 0.410816i \(0.134756\pi\)
\(18\) 8453.98 0.341671
\(19\) −31065.7 −1.03907 −0.519533 0.854451i \(-0.673894\pi\)
−0.519533 + 0.854451i \(0.673894\pi\)
\(20\) −2600.38 −0.0726829
\(21\) −5766.47 −0.135876
\(22\) −57683.5 −1.15497
\(23\) 52594.0 0.901341 0.450670 0.892690i \(-0.351185\pi\)
0.450670 + 0.892690i \(0.351185\pi\)
\(24\) 38048.2 0.561817
\(25\) 82759.6 1.05932
\(26\) −146697. −1.63688
\(27\) −19683.0 −0.192450
\(28\) 1384.61 0.0119199
\(29\) −209912. −1.59825 −0.799123 0.601167i \(-0.794702\pi\)
−0.799123 + 0.601167i \(0.794702\pi\)
\(30\) 125590. 0.849240
\(31\) −11865.2 −0.0715333 −0.0357667 0.999360i \(-0.511387\pi\)
−0.0357667 + 0.999360i \(0.511387\pi\)
\(32\) −18759.2 −0.101202
\(33\) 134302. 0.650553
\(34\) 428347. 1.86904
\(35\) −85665.0 −0.337727
\(36\) 4726.15 0.0168830
\(37\) 16919.7 0.0549143 0.0274572 0.999623i \(-0.491259\pi\)
0.0274572 + 0.999623i \(0.491259\pi\)
\(38\) −360259. −1.06505
\(39\) 341549. 0.921990
\(40\) 565233. 1.39643
\(41\) 801040. 1.81514 0.907571 0.419898i \(-0.137934\pi\)
0.907571 + 0.419898i \(0.137934\pi\)
\(42\) −66871.9 −0.139274
\(43\) 720238. 1.38145 0.690727 0.723115i \(-0.257290\pi\)
0.690727 + 0.723115i \(0.257290\pi\)
\(44\) −32247.7 −0.0570708
\(45\) −292405. −0.478345
\(46\) 609916. 0.923885
\(47\) −54403.0 −0.0764329 −0.0382164 0.999269i \(-0.512168\pi\)
−0.0382164 + 0.999269i \(0.512168\pi\)
\(48\) 463639. 0.605111
\(49\) −777930. −0.944613
\(50\) 959737. 1.08582
\(51\) −997299. −1.05276
\(52\) −82010.4 −0.0808830
\(53\) −295498. −0.272639 −0.136320 0.990665i \(-0.543527\pi\)
−0.136320 + 0.990665i \(0.543527\pi\)
\(54\) −228258. −0.197264
\(55\) 1.99515e6 1.61698
\(56\) −300965. −0.229012
\(57\) 838773. 0.599905
\(58\) −2.43428e6 −1.63822
\(59\) −205379. −0.130189
\(60\) 70210.3 0.0419635
\(61\) 1.37027e6 0.772952 0.386476 0.922300i \(-0.373692\pi\)
0.386476 + 0.922300i \(0.373692\pi\)
\(62\) −137597. −0.0733225
\(63\) 155695. 0.0784480
\(64\) 1.98045e6 0.944350
\(65\) 5.07395e6 2.29166
\(66\) 1.55746e6 0.666825
\(67\) 3.05207e6 1.23975 0.619873 0.784702i \(-0.287184\pi\)
0.619873 + 0.784702i \(0.287184\pi\)
\(68\) 239465. 0.0923551
\(69\) −1.42004e6 −0.520389
\(70\) −993430. −0.346174
\(71\) −4.70310e6 −1.55948 −0.779740 0.626104i \(-0.784649\pi\)
−0.779740 + 0.626104i \(0.784649\pi\)
\(72\) −1.02730e6 −0.324365
\(73\) −3.65576e6 −1.09989 −0.549943 0.835202i \(-0.685351\pi\)
−0.549943 + 0.835202i \(0.685351\pi\)
\(74\) 196212. 0.0562878
\(75\) −2.23451e6 −0.611600
\(76\) −201401. −0.0526275
\(77\) −1.06234e6 −0.265184
\(78\) 3.96083e6 0.945051
\(79\) 6.37313e6 1.45431 0.727157 0.686471i \(-0.240841\pi\)
0.727157 + 0.686471i \(0.240841\pi\)
\(80\) 6.88768e6 1.50404
\(81\) 531441. 0.111111
\(82\) 9.28941e6 1.86054
\(83\) −2.08701e6 −0.400636 −0.200318 0.979731i \(-0.564198\pi\)
−0.200318 + 0.979731i \(0.564198\pi\)
\(84\) −37384.4 −0.00688197
\(85\) −1.48156e7 −2.61669
\(86\) 8.35238e6 1.41601
\(87\) 5.66762e6 0.922748
\(88\) 7.00953e6 1.09648
\(89\) 1.43644e6 0.215984 0.107992 0.994152i \(-0.465558\pi\)
0.107992 + 0.994152i \(0.465558\pi\)
\(90\) −3.39093e6 −0.490309
\(91\) −2.70169e6 −0.375829
\(92\) 340970. 0.0456519
\(93\) 320360. 0.0412998
\(94\) −630894. −0.0783446
\(95\) 1.24606e7 1.49109
\(96\) 506498. 0.0584290
\(97\) −6.42693e6 −0.714994 −0.357497 0.933914i \(-0.616370\pi\)
−0.357497 + 0.933914i \(0.616370\pi\)
\(98\) −9.02140e6 −0.968240
\(99\) −3.62615e6 −0.375597
\(100\) 536536. 0.0536536
\(101\) 1.06724e6 0.103071 0.0515356 0.998671i \(-0.483588\pi\)
0.0515356 + 0.998671i \(0.483588\pi\)
\(102\) −1.15654e7 −1.07909
\(103\) −9.80605e6 −0.884227 −0.442113 0.896959i \(-0.645771\pi\)
−0.442113 + 0.896959i \(0.645771\pi\)
\(104\) 1.78262e7 1.55397
\(105\) 2.31295e6 0.194987
\(106\) −3.42679e6 −0.279459
\(107\) −1.96135e7 −1.54779 −0.773895 0.633314i \(-0.781694\pi\)
−0.773895 + 0.633314i \(0.781694\pi\)
\(108\) −127606. −0.00974739
\(109\) 2.55432e7 1.88922 0.944610 0.328194i \(-0.106440\pi\)
0.944610 + 0.328194i \(0.106440\pi\)
\(110\) 2.31371e7 1.65743
\(111\) −456831. −0.0317048
\(112\) −3.66743e6 −0.246660
\(113\) −1.24785e7 −0.813559 −0.406780 0.913526i \(-0.633348\pi\)
−0.406780 + 0.913526i \(0.633348\pi\)
\(114\) 9.72698e6 0.614909
\(115\) −2.10957e7 −1.29345
\(116\) −1.36087e6 −0.0809494
\(117\) −9.22181e6 −0.532311
\(118\) −2.38172e6 −0.133445
\(119\) 7.88875e6 0.429135
\(120\) −1.52613e7 −0.806227
\(121\) 5.25490e6 0.269659
\(122\) 1.58906e7 0.792284
\(123\) −2.16281e7 −1.04797
\(124\) −76922.7 −0.00362308
\(125\) −1.85896e6 −0.0851305
\(126\) 1.80554e6 0.0804101
\(127\) −1.73455e7 −0.751403 −0.375701 0.926741i \(-0.622598\pi\)
−0.375701 + 0.926741i \(0.622598\pi\)
\(128\) 2.53678e7 1.06917
\(129\) −1.94464e7 −0.797583
\(130\) 5.88410e7 2.34897
\(131\) −1.36249e7 −0.529520 −0.264760 0.964314i \(-0.585293\pi\)
−0.264760 + 0.964314i \(0.585293\pi\)
\(132\) 870687. 0.0329498
\(133\) −6.63478e6 −0.244538
\(134\) 3.53939e7 1.27075
\(135\) 7.89493e6 0.276172
\(136\) −5.20514e7 −1.77438
\(137\) −1.67308e7 −0.555898 −0.277949 0.960596i \(-0.589655\pi\)
−0.277949 + 0.960596i \(0.589655\pi\)
\(138\) −1.64677e7 −0.533405
\(139\) 1.17094e7 0.369813 0.184907 0.982756i \(-0.440802\pi\)
0.184907 + 0.982756i \(0.440802\pi\)
\(140\) −555371. −0.0171055
\(141\) 1.46888e6 0.0441285
\(142\) −5.45403e7 −1.59848
\(143\) 6.29226e7 1.79941
\(144\) −1.25182e7 −0.349361
\(145\) 8.41965e7 2.29354
\(146\) −4.23947e7 −1.12740
\(147\) 2.10041e7 0.545373
\(148\) 109691. 0.00278135
\(149\) 153329. 0.00379728 0.00189864 0.999998i \(-0.499396\pi\)
0.00189864 + 0.999998i \(0.499396\pi\)
\(150\) −2.59129e7 −0.626898
\(151\) 7.64791e7 1.80769 0.903843 0.427863i \(-0.140734\pi\)
0.903843 + 0.427863i \(0.140734\pi\)
\(152\) 4.37775e7 1.01111
\(153\) 2.69271e7 0.607812
\(154\) −1.23196e7 −0.271816
\(155\) 4.75917e6 0.102653
\(156\) 2.21428e6 0.0466978
\(157\) −6.86729e6 −0.141624 −0.0708119 0.997490i \(-0.522559\pi\)
−0.0708119 + 0.997490i \(0.522559\pi\)
\(158\) 7.39072e7 1.49069
\(159\) 7.97844e6 0.157408
\(160\) 7.52439e6 0.145228
\(161\) 1.12327e7 0.212125
\(162\) 6.16295e6 0.113890
\(163\) −1.89828e7 −0.343323 −0.171662 0.985156i \(-0.554914\pi\)
−0.171662 + 0.985156i \(0.554914\pi\)
\(164\) 5.19319e6 0.0919350
\(165\) −5.38690e7 −0.933566
\(166\) −2.42024e7 −0.410657
\(167\) −1.70383e7 −0.283085 −0.141543 0.989932i \(-0.545206\pi\)
−0.141543 + 0.989932i \(0.545206\pi\)
\(168\) 8.12607e6 0.132220
\(169\) 9.72727e7 1.55020
\(170\) −1.71812e8 −2.68214
\(171\) −2.26469e7 −0.346355
\(172\) 4.66935e6 0.0699692
\(173\) −8.07641e6 −0.118592 −0.0592962 0.998240i \(-0.518886\pi\)
−0.0592962 + 0.998240i \(0.518886\pi\)
\(174\) 6.57255e7 0.945827
\(175\) 1.76752e7 0.249305
\(176\) 8.54149e7 1.18097
\(177\) 5.54523e6 0.0751646
\(178\) 1.66579e7 0.221386
\(179\) 4.99947e7 0.651536 0.325768 0.945450i \(-0.394377\pi\)
0.325768 + 0.945450i \(0.394377\pi\)
\(180\) −1.89568e6 −0.0242276
\(181\) 7.07416e7 0.886747 0.443373 0.896337i \(-0.353782\pi\)
0.443373 + 0.896337i \(0.353782\pi\)
\(182\) −3.13306e7 −0.385229
\(183\) −3.69973e7 −0.446264
\(184\) −7.41151e7 −0.877091
\(185\) −6.78655e6 −0.0788039
\(186\) 3.71511e6 0.0423327
\(187\) −1.83730e8 −2.05463
\(188\) −352698. −0.00387124
\(189\) −4.20375e6 −0.0452920
\(190\) 1.44501e8 1.52839
\(191\) −1.40226e8 −1.45617 −0.728086 0.685486i \(-0.759590\pi\)
−0.728086 + 0.685486i \(0.759590\pi\)
\(192\) −5.34721e7 −0.545221
\(193\) −5.81729e7 −0.582465 −0.291233 0.956652i \(-0.594065\pi\)
−0.291233 + 0.956652i \(0.594065\pi\)
\(194\) −7.45311e7 −0.732878
\(195\) −1.36997e8 −1.32309
\(196\) −5.04337e6 −0.0478436
\(197\) 6.06171e7 0.564889 0.282445 0.959284i \(-0.408855\pi\)
0.282445 + 0.959284i \(0.408855\pi\)
\(198\) −4.20513e7 −0.384992
\(199\) 234342. 0.00210797 0.00105398 0.999999i \(-0.499665\pi\)
0.00105398 + 0.999999i \(0.499665\pi\)
\(200\) −1.16624e8 −1.03082
\(201\) −8.24059e7 −0.715768
\(202\) 1.23765e7 0.105649
\(203\) −4.48314e7 −0.376137
\(204\) −6.46555e6 −0.0533212
\(205\) −3.21300e8 −2.60479
\(206\) −1.13718e8 −0.906343
\(207\) 3.83410e7 0.300447
\(208\) 2.17222e8 1.67372
\(209\) 1.54525e8 1.17081
\(210\) 2.68226e7 0.199863
\(211\) −2.04520e8 −1.49881 −0.749406 0.662111i \(-0.769661\pi\)
−0.749406 + 0.662111i \(0.769661\pi\)
\(212\) −1.91573e6 −0.0138089
\(213\) 1.26984e8 0.900366
\(214\) −2.27452e8 −1.58650
\(215\) −2.88891e8 −1.98243
\(216\) 2.77372e7 0.187272
\(217\) −2.53408e6 −0.0168349
\(218\) 2.96216e8 1.93647
\(219\) 9.87055e7 0.635019
\(220\) 1.29347e7 0.0818985
\(221\) −4.67251e8 −2.91191
\(222\) −5.29772e6 −0.0324978
\(223\) 2.44062e8 1.47378 0.736892 0.676011i \(-0.236293\pi\)
0.736892 + 0.676011i \(0.236293\pi\)
\(224\) −4.00645e6 −0.0238173
\(225\) 6.03318e7 0.353108
\(226\) −1.44710e8 −0.833908
\(227\) −1.77415e8 −1.00670 −0.503350 0.864083i \(-0.667899\pi\)
−0.503350 + 0.864083i \(0.667899\pi\)
\(228\) 5.43781e6 0.0303845
\(229\) 2.17580e8 1.19728 0.598638 0.801020i \(-0.295709\pi\)
0.598638 + 0.801020i \(0.295709\pi\)
\(230\) −2.44640e8 −1.32581
\(231\) 2.86832e7 0.153104
\(232\) 2.95806e8 1.55525
\(233\) −3.27814e8 −1.69778 −0.848890 0.528570i \(-0.822728\pi\)
−0.848890 + 0.528570i \(0.822728\pi\)
\(234\) −1.06942e8 −0.545625
\(235\) 2.18213e7 0.109684
\(236\) −1.33148e6 −0.00659393
\(237\) −1.72075e8 −0.839648
\(238\) 9.14833e7 0.439868
\(239\) 2.79462e8 1.32413 0.662064 0.749448i \(-0.269681\pi\)
0.662064 + 0.749448i \(0.269681\pi\)
\(240\) −1.85967e8 −0.868355
\(241\) 3.83280e8 1.76383 0.881915 0.471409i \(-0.156254\pi\)
0.881915 + 0.471409i \(0.156254\pi\)
\(242\) 6.09394e7 0.276404
\(243\) −1.43489e7 −0.0641500
\(244\) 8.88356e6 0.0391492
\(245\) 3.12031e8 1.35555
\(246\) −2.50814e8 −1.07418
\(247\) 3.92979e8 1.65932
\(248\) 1.67203e7 0.0696088
\(249\) 5.63492e7 0.231308
\(250\) −2.15578e7 −0.0872598
\(251\) −4.23948e8 −1.69221 −0.846106 0.533014i \(-0.821059\pi\)
−0.846106 + 0.533014i \(0.821059\pi\)
\(252\) 1.00938e6 0.00397330
\(253\) −2.61610e8 −1.01562
\(254\) −2.01150e8 −0.770196
\(255\) 4.00021e8 1.51075
\(256\) 4.06850e7 0.151564
\(257\) 1.80864e8 0.664638 0.332319 0.943167i \(-0.392169\pi\)
0.332319 + 0.943167i \(0.392169\pi\)
\(258\) −2.25514e8 −0.817532
\(259\) 3.61358e6 0.0129237
\(260\) 3.28947e7 0.116070
\(261\) −1.53026e8 −0.532749
\(262\) −1.58003e8 −0.542764
\(263\) 4.23691e8 1.43616 0.718082 0.695958i \(-0.245020\pi\)
0.718082 + 0.695958i \(0.245020\pi\)
\(264\) −1.89257e8 −0.633051
\(265\) 1.18525e8 0.391247
\(266\) −7.69415e7 −0.250654
\(267\) −3.87838e7 −0.124699
\(268\) 1.97868e7 0.0627918
\(269\) 3.48418e8 1.09136 0.545679 0.837994i \(-0.316272\pi\)
0.545679 + 0.837994i \(0.316272\pi\)
\(270\) 9.15551e7 0.283080
\(271\) −1.26441e8 −0.385918 −0.192959 0.981207i \(-0.561809\pi\)
−0.192959 + 0.981207i \(0.561809\pi\)
\(272\) −6.34275e8 −1.91111
\(273\) 7.29455e7 0.216985
\(274\) −1.94022e8 −0.569802
\(275\) −4.11658e8 −1.19364
\(276\) −9.20620e6 −0.0263571
\(277\) −2.22094e8 −0.627852 −0.313926 0.949447i \(-0.601644\pi\)
−0.313926 + 0.949447i \(0.601644\pi\)
\(278\) 1.35790e8 0.379063
\(279\) −8.64971e6 −0.0238444
\(280\) 1.20719e8 0.328640
\(281\) −3.20287e8 −0.861128 −0.430564 0.902560i \(-0.641685\pi\)
−0.430564 + 0.902560i \(0.641685\pi\)
\(282\) 1.70341e7 0.0452323
\(283\) 2.46262e8 0.645870 0.322935 0.946421i \(-0.395331\pi\)
0.322935 + 0.946421i \(0.395331\pi\)
\(284\) −3.04905e7 −0.0789860
\(285\) −3.36435e8 −0.860884
\(286\) 7.29694e8 1.84442
\(287\) 1.71080e8 0.427183
\(288\) −1.36754e7 −0.0337340
\(289\) 9.54005e8 2.32492
\(290\) 9.76400e8 2.35090
\(291\) 1.73527e8 0.412802
\(292\) −2.37005e7 −0.0557080
\(293\) 2.26005e8 0.524905 0.262453 0.964945i \(-0.415469\pi\)
0.262453 + 0.964945i \(0.415469\pi\)
\(294\) 2.43578e8 0.559013
\(295\) 8.23784e7 0.186826
\(296\) −2.38431e7 −0.0534369
\(297\) 9.79060e7 0.216851
\(298\) 1.77811e6 0.00389226
\(299\) −6.65312e8 −1.43938
\(300\) −1.44865e7 −0.0309769
\(301\) 1.53823e8 0.325117
\(302\) 8.86903e8 1.85290
\(303\) −2.88155e7 −0.0595082
\(304\) 5.33453e8 1.08903
\(305\) −5.49622e8 −1.10921
\(306\) 3.12265e8 0.623015
\(307\) 5.53693e8 1.09216 0.546078 0.837734i \(-0.316120\pi\)
0.546078 + 0.837734i \(0.316120\pi\)
\(308\) −6.88722e6 −0.0134313
\(309\) 2.64763e8 0.510509
\(310\) 5.51906e7 0.105220
\(311\) 8.72060e8 1.64394 0.821968 0.569534i \(-0.192876\pi\)
0.821968 + 0.569534i \(0.192876\pi\)
\(312\) −4.81308e8 −0.897185
\(313\) 3.27729e8 0.604102 0.302051 0.953292i \(-0.402329\pi\)
0.302051 + 0.953292i \(0.402329\pi\)
\(314\) −7.96378e7 −0.145166
\(315\) −6.24498e7 −0.112576
\(316\) 4.13174e7 0.0736594
\(317\) 1.07872e9 1.90196 0.950982 0.309246i \(-0.100077\pi\)
0.950982 + 0.309246i \(0.100077\pi\)
\(318\) 9.25234e7 0.161345
\(319\) 1.04413e9 1.80089
\(320\) −7.94365e8 −1.35518
\(321\) 5.29564e8 0.893616
\(322\) 1.30262e8 0.217431
\(323\) −1.14747e9 −1.89467
\(324\) 3.44536e6 0.00562766
\(325\) −1.04690e9 −1.69167
\(326\) −2.20137e8 −0.351910
\(327\) −6.89667e8 −1.09074
\(328\) −1.12882e9 −1.76631
\(329\) −1.16190e7 −0.0179880
\(330\) −6.24702e8 −0.956916
\(331\) 1.91834e8 0.290755 0.145378 0.989376i \(-0.453560\pi\)
0.145378 + 0.989376i \(0.453560\pi\)
\(332\) −1.35302e7 −0.0202918
\(333\) 1.23344e7 0.0183048
\(334\) −1.97587e8 −0.290166
\(335\) −1.22420e9 −1.77908
\(336\) 9.90206e7 0.142409
\(337\) 4.73052e8 0.673294 0.336647 0.941631i \(-0.390707\pi\)
0.336647 + 0.941631i \(0.390707\pi\)
\(338\) 1.12804e9 1.58897
\(339\) 3.36921e8 0.469709
\(340\) −9.60504e7 −0.132533
\(341\) 5.90190e7 0.0806031
\(342\) −2.62628e8 −0.355018
\(343\) −3.42031e8 −0.457653
\(344\) −1.01496e9 −1.34429
\(345\) 5.69583e8 0.746776
\(346\) −9.36596e7 −0.121559
\(347\) −2.39433e8 −0.307631 −0.153815 0.988100i \(-0.549156\pi\)
−0.153815 + 0.988100i \(0.549156\pi\)
\(348\) 3.67435e7 0.0467362
\(349\) 2.55293e8 0.321477 0.160738 0.986997i \(-0.448612\pi\)
0.160738 + 0.986997i \(0.448612\pi\)
\(350\) 2.04974e8 0.255541
\(351\) 2.48989e8 0.307330
\(352\) 9.33108e7 0.114034
\(353\) 9.16807e8 1.10934 0.554672 0.832069i \(-0.312844\pi\)
0.554672 + 0.832069i \(0.312844\pi\)
\(354\) 6.43063e7 0.0770446
\(355\) 1.88643e9 2.23791
\(356\) 9.31252e6 0.0109394
\(357\) −2.12996e8 −0.247761
\(358\) 5.79773e8 0.667832
\(359\) −1.06121e8 −0.121051 −0.0605256 0.998167i \(-0.519278\pi\)
−0.0605256 + 0.998167i \(0.519278\pi\)
\(360\) 4.12055e8 0.465475
\(361\) 7.12032e7 0.0796570
\(362\) 8.20368e8 0.908926
\(363\) −1.41882e8 −0.155688
\(364\) −1.75152e7 −0.0190353
\(365\) 1.46634e9 1.57837
\(366\) −4.29046e8 −0.457426
\(367\) 1.13340e9 1.19689 0.598444 0.801164i \(-0.295786\pi\)
0.598444 + 0.801164i \(0.295786\pi\)
\(368\) −9.03134e8 −0.944680
\(369\) 5.83958e8 0.605047
\(370\) −7.87014e7 −0.0807749
\(371\) −6.31103e7 −0.0641640
\(372\) 2.07691e6 0.00209179
\(373\) 3.48649e8 0.347863 0.173931 0.984758i \(-0.444353\pi\)
0.173931 + 0.984758i \(0.444353\pi\)
\(374\) −2.13066e9 −2.10602
\(375\) 5.01920e7 0.0491501
\(376\) 7.66643e7 0.0743765
\(377\) 2.65537e9 2.55229
\(378\) −4.87496e7 −0.0464248
\(379\) −1.97757e8 −0.186592 −0.0932962 0.995638i \(-0.529740\pi\)
−0.0932962 + 0.995638i \(0.529740\pi\)
\(380\) 8.07826e7 0.0755223
\(381\) 4.68327e8 0.433822
\(382\) −1.62616e9 −1.49259
\(383\) −1.60320e9 −1.45811 −0.729056 0.684454i \(-0.760041\pi\)
−0.729056 + 0.684454i \(0.760041\pi\)
\(384\) −6.84930e8 −0.617287
\(385\) 4.26110e8 0.380547
\(386\) −6.74613e8 −0.597034
\(387\) 5.25054e8 0.460485
\(388\) −4.16662e7 −0.0362137
\(389\) −7.74392e8 −0.667018 −0.333509 0.942747i \(-0.608233\pi\)
−0.333509 + 0.942747i \(0.608233\pi\)
\(390\) −1.58871e9 −1.35618
\(391\) 1.94267e9 1.64354
\(392\) 1.09625e9 0.919199
\(393\) 3.67871e8 0.305718
\(394\) 7.02957e8 0.579018
\(395\) −2.55629e9 −2.08699
\(396\) −2.35085e7 −0.0190236
\(397\) −1.24932e9 −1.00209 −0.501044 0.865422i \(-0.667050\pi\)
−0.501044 + 0.865422i \(0.667050\pi\)
\(398\) 2.71759e6 0.00216069
\(399\) 1.79139e8 0.141184
\(400\) −1.42113e9 −1.11026
\(401\) −2.10043e9 −1.62668 −0.813341 0.581788i \(-0.802353\pi\)
−0.813341 + 0.581788i \(0.802353\pi\)
\(402\) −9.55635e8 −0.733670
\(403\) 1.50094e8 0.114234
\(404\) 6.91899e6 0.00522045
\(405\) −2.13163e8 −0.159448
\(406\) −5.19896e8 −0.385545
\(407\) −8.41607e7 −0.0618770
\(408\) 1.40539e9 1.02444
\(409\) 2.09675e9 1.51535 0.757677 0.652629i \(-0.226334\pi\)
0.757677 + 0.652629i \(0.226334\pi\)
\(410\) −3.72602e9 −2.66994
\(411\) 4.51732e8 0.320948
\(412\) −6.35732e7 −0.0447851
\(413\) −4.38634e7 −0.0306392
\(414\) 4.44629e8 0.307962
\(415\) 8.37107e8 0.574927
\(416\) 2.37303e8 0.161613
\(417\) −3.16153e8 −0.213512
\(418\) 1.79198e9 1.20009
\(419\) −7.54525e8 −0.501100 −0.250550 0.968104i \(-0.580612\pi\)
−0.250550 + 0.968104i \(0.580612\pi\)
\(420\) 1.49950e7 0.00987585
\(421\) −1.83091e8 −0.119586 −0.0597931 0.998211i \(-0.519044\pi\)
−0.0597931 + 0.998211i \(0.519044\pi\)
\(422\) −2.37175e9 −1.53630
\(423\) −3.96598e7 −0.0254776
\(424\) 4.16413e8 0.265304
\(425\) 3.05689e9 1.93161
\(426\) 1.47259e9 0.922886
\(427\) 2.92653e8 0.181909
\(428\) −1.27155e8 −0.0783938
\(429\) −1.69891e9 −1.03889
\(430\) −3.35017e9 −2.03202
\(431\) 2.72495e8 0.163941 0.0819706 0.996635i \(-0.473879\pi\)
0.0819706 + 0.996635i \(0.473879\pi\)
\(432\) 3.37993e8 0.201704
\(433\) −9.06889e8 −0.536842 −0.268421 0.963302i \(-0.586502\pi\)
−0.268421 + 0.963302i \(0.586502\pi\)
\(434\) −2.93869e7 −0.0172560
\(435\) −2.27330e9 −1.32417
\(436\) 1.65598e8 0.0956870
\(437\) −1.63387e9 −0.936552
\(438\) 1.14466e9 0.650902
\(439\) −6.54375e7 −0.0369148 −0.0184574 0.999830i \(-0.505876\pi\)
−0.0184574 + 0.999830i \(0.505876\pi\)
\(440\) −2.81155e9 −1.57348
\(441\) −5.67111e8 −0.314871
\(442\) −5.41857e9 −2.98474
\(443\) 2.60048e9 1.42115 0.710575 0.703621i \(-0.248435\pi\)
0.710575 + 0.703621i \(0.248435\pi\)
\(444\) −2.96166e6 −0.00160581
\(445\) −5.76162e8 −0.309945
\(446\) 2.83031e9 1.51065
\(447\) −4.13989e6 −0.00219236
\(448\) 4.22970e8 0.222247
\(449\) 2.52865e9 1.31834 0.659168 0.751996i \(-0.270909\pi\)
0.659168 + 0.751996i \(0.270909\pi\)
\(450\) 6.99648e8 0.361939
\(451\) −3.98448e9 −2.04529
\(452\) −8.08992e7 −0.0412059
\(453\) −2.06493e9 −1.04367
\(454\) −2.05742e9 −1.03188
\(455\) 1.08366e9 0.539327
\(456\) −1.18199e9 −0.583765
\(457\) −1.26888e9 −0.621888 −0.310944 0.950428i \(-0.600645\pi\)
−0.310944 + 0.950428i \(0.600645\pi\)
\(458\) 2.52320e9 1.22722
\(459\) −7.27031e8 −0.350920
\(460\) −1.36765e8 −0.0655121
\(461\) 6.17722e8 0.293656 0.146828 0.989162i \(-0.453094\pi\)
0.146828 + 0.989162i \(0.453094\pi\)
\(462\) 3.32630e8 0.156933
\(463\) 1.76885e9 0.828241 0.414120 0.910222i \(-0.364089\pi\)
0.414120 + 0.910222i \(0.364089\pi\)
\(464\) 3.60456e9 1.67510
\(465\) −1.28498e8 −0.0592666
\(466\) −3.80155e9 −1.74024
\(467\) 2.09197e9 0.950489 0.475244 0.879854i \(-0.342360\pi\)
0.475244 + 0.879854i \(0.342360\pi\)
\(468\) −5.97856e7 −0.0269610
\(469\) 6.51839e8 0.291767
\(470\) 2.53054e8 0.112427
\(471\) 1.85417e8 0.0817666
\(472\) 2.89419e8 0.126686
\(473\) −3.58257e9 −1.55661
\(474\) −1.99549e9 −0.860649
\(475\) −2.57098e9 −1.10071
\(476\) 5.11432e7 0.0217352
\(477\) −2.15418e8 −0.0908798
\(478\) 3.24083e9 1.35725
\(479\) −7.21973e8 −0.300156 −0.150078 0.988674i \(-0.547952\pi\)
−0.150078 + 0.988674i \(0.547952\pi\)
\(480\) −2.03158e8 −0.0838476
\(481\) −2.14033e8 −0.0876945
\(482\) 4.44478e9 1.80795
\(483\) −3.03282e8 −0.122470
\(484\) 3.40678e7 0.0136580
\(485\) 2.57787e9 1.02604
\(486\) −1.66400e8 −0.0657545
\(487\) 2.53074e9 0.992878 0.496439 0.868072i \(-0.334641\pi\)
0.496439 + 0.868072i \(0.334641\pi\)
\(488\) −1.93098e9 −0.752156
\(489\) 5.12535e8 0.198218
\(490\) 3.61852e9 1.38946
\(491\) 4.02112e9 1.53307 0.766535 0.642203i \(-0.221979\pi\)
0.766535 + 0.642203i \(0.221979\pi\)
\(492\) −1.40216e8 −0.0530787
\(493\) −7.75351e9 −2.91430
\(494\) 4.55725e9 1.70082
\(495\) 1.45446e9 0.538995
\(496\) 2.03746e8 0.0749729
\(497\) −1.00445e9 −0.367014
\(498\) 6.53463e8 0.237093
\(499\) −1.21528e9 −0.437849 −0.218924 0.975742i \(-0.570255\pi\)
−0.218924 + 0.975742i \(0.570255\pi\)
\(500\) −1.20518e7 −0.00431177
\(501\) 4.60033e8 0.163439
\(502\) −4.91640e9 −1.73454
\(503\) 4.23068e9 1.48225 0.741126 0.671366i \(-0.234292\pi\)
0.741126 + 0.671366i \(0.234292\pi\)
\(504\) −2.19404e8 −0.0763374
\(505\) −4.28075e8 −0.147911
\(506\) −3.03381e9 −1.04103
\(507\) −2.62636e9 −0.895008
\(508\) −1.12452e8 −0.0380577
\(509\) 5.89664e9 1.98195 0.990974 0.134052i \(-0.0427989\pi\)
0.990974 + 0.134052i \(0.0427989\pi\)
\(510\) 4.63892e9 1.54853
\(511\) −7.80771e8 −0.258851
\(512\) −2.77526e9 −0.913818
\(513\) 6.11465e8 0.199968
\(514\) 2.09742e9 0.681262
\(515\) 3.93325e9 1.26890
\(516\) −1.26072e8 −0.0403967
\(517\) 2.70608e8 0.0861239
\(518\) 4.19055e7 0.0132470
\(519\) 2.18063e8 0.0684694
\(520\) −7.15017e9 −2.23000
\(521\) −3.53847e8 −0.109618 −0.0548092 0.998497i \(-0.517455\pi\)
−0.0548092 + 0.998497i \(0.517455\pi\)
\(522\) −1.77459e9 −0.546074
\(523\) 1.49203e9 0.456059 0.228029 0.973654i \(-0.426772\pi\)
0.228029 + 0.973654i \(0.426772\pi\)
\(524\) −8.83307e7 −0.0268196
\(525\) −4.77231e8 −0.143936
\(526\) 4.91341e9 1.47209
\(527\) −4.38264e8 −0.130436
\(528\) −2.30620e9 −0.681834
\(529\) −6.38695e8 −0.187585
\(530\) 1.37450e9 0.401033
\(531\) −1.49721e8 −0.0433963
\(532\) −4.30137e7 −0.0123856
\(533\) −1.01331e10 −2.89866
\(534\) −4.49764e8 −0.127817
\(535\) 7.86706e9 2.22113
\(536\) −4.30096e9 −1.20639
\(537\) −1.34986e9 −0.376164
\(538\) 4.04049e9 1.11866
\(539\) 3.86953e9 1.06438
\(540\) 5.11833e7 0.0139878
\(541\) −2.13831e9 −0.580603 −0.290302 0.956935i \(-0.593756\pi\)
−0.290302 + 0.956935i \(0.593756\pi\)
\(542\) −1.46630e9 −0.395571
\(543\) −1.91002e9 −0.511964
\(544\) −6.92908e8 −0.184535
\(545\) −1.02455e10 −2.71110
\(546\) 8.45926e8 0.222412
\(547\) −2.11049e9 −0.551351 −0.275676 0.961251i \(-0.588902\pi\)
−0.275676 + 0.961251i \(0.588902\pi\)
\(548\) −1.08467e8 −0.0281556
\(549\) 9.98928e8 0.257651
\(550\) −4.77387e9 −1.22349
\(551\) 6.52104e9 1.66068
\(552\) 2.00111e9 0.506389
\(553\) 1.36113e9 0.342264
\(554\) −2.57555e9 −0.643556
\(555\) 1.83237e8 0.0454975
\(556\) 7.59126e7 0.0187306
\(557\) −5.51882e9 −1.35317 −0.676586 0.736364i \(-0.736541\pi\)
−0.676586 + 0.736364i \(0.736541\pi\)
\(558\) −1.00308e8 −0.0244408
\(559\) −9.11098e9 −2.20609
\(560\) 1.47102e9 0.353966
\(561\) 4.96071e9 1.18624
\(562\) −3.71427e9 −0.882666
\(563\) −3.94126e8 −0.0930798 −0.0465399 0.998916i \(-0.514819\pi\)
−0.0465399 + 0.998916i \(0.514819\pi\)
\(564\) 9.52284e6 0.00223506
\(565\) 5.00520e9 1.16749
\(566\) 2.85582e9 0.662024
\(567\) 1.13501e8 0.0261493
\(568\) 6.62757e9 1.51752
\(569\) −5.23096e9 −1.19039 −0.595194 0.803582i \(-0.702925\pi\)
−0.595194 + 0.803582i \(0.702925\pi\)
\(570\) −3.90153e9 −0.882416
\(571\) 3.54290e9 0.796401 0.398201 0.917298i \(-0.369635\pi\)
0.398201 + 0.917298i \(0.369635\pi\)
\(572\) 4.07931e8 0.0911383
\(573\) 3.78611e9 0.840721
\(574\) 1.98397e9 0.437867
\(575\) 4.35266e9 0.954811
\(576\) 1.44375e9 0.314783
\(577\) −5.36264e8 −0.116215 −0.0581077 0.998310i \(-0.518507\pi\)
−0.0581077 + 0.998310i \(0.518507\pi\)
\(578\) 1.10633e10 2.38307
\(579\) 1.57067e9 0.336286
\(580\) 5.45851e8 0.116165
\(581\) −4.45728e8 −0.0942873
\(582\) 2.01234e9 0.423127
\(583\) 1.46985e9 0.307208
\(584\) 5.15167e9 1.07029
\(585\) 3.69891e9 0.763885
\(586\) 2.62090e9 0.538034
\(587\) −1.65503e9 −0.337733 −0.168867 0.985639i \(-0.554011\pi\)
−0.168867 + 0.985639i \(0.554011\pi\)
\(588\) 1.36171e8 0.0276225
\(589\) 3.68599e8 0.0743278
\(590\) 9.55316e8 0.191498
\(591\) −1.63666e9 −0.326139
\(592\) −2.90541e8 −0.0575548
\(593\) −8.05027e9 −1.58533 −0.792664 0.609659i \(-0.791307\pi\)
−0.792664 + 0.609659i \(0.791307\pi\)
\(594\) 1.13538e9 0.222275
\(595\) −3.16421e9 −0.615823
\(596\) 994042. 0.000192328 0
\(597\) −6.32723e6 −0.00121704
\(598\) −7.71541e9 −1.47538
\(599\) −2.91923e7 −0.00554976 −0.00277488 0.999996i \(-0.500883\pi\)
−0.00277488 + 0.999996i \(0.500883\pi\)
\(600\) 3.14886e9 0.595146
\(601\) −5.20322e9 −0.977713 −0.488856 0.872364i \(-0.662586\pi\)
−0.488856 + 0.872364i \(0.662586\pi\)
\(602\) 1.78384e9 0.333249
\(603\) 2.22496e9 0.413249
\(604\) 4.95818e8 0.0915574
\(605\) −2.10776e9 −0.386970
\(606\) −3.34164e8 −0.0609966
\(607\) 7.88385e9 1.43080 0.715398 0.698717i \(-0.246245\pi\)
0.715398 + 0.698717i \(0.246245\pi\)
\(608\) 5.82766e8 0.105155
\(609\) 1.21045e9 0.217163
\(610\) −6.37379e9 −1.13696
\(611\) 6.88195e8 0.122058
\(612\) 1.74570e8 0.0307850
\(613\) −3.27886e9 −0.574926 −0.287463 0.957792i \(-0.592812\pi\)
−0.287463 + 0.957792i \(0.592812\pi\)
\(614\) 6.42100e9 1.11947
\(615\) 8.67511e9 1.50388
\(616\) 1.49704e9 0.258049
\(617\) 1.20684e9 0.206848 0.103424 0.994637i \(-0.467020\pi\)
0.103424 + 0.994637i \(0.467020\pi\)
\(618\) 3.07038e9 0.523277
\(619\) −4.26264e9 −0.722372 −0.361186 0.932494i \(-0.617628\pi\)
−0.361186 + 0.932494i \(0.617628\pi\)
\(620\) 3.08540e7 0.00519925
\(621\) −1.03521e9 −0.173463
\(622\) 1.01130e10 1.68505
\(623\) 3.06784e8 0.0508306
\(624\) −5.86501e9 −0.966323
\(625\) −5.71996e9 −0.937158
\(626\) 3.80057e9 0.619211
\(627\) −4.17217e9 −0.675968
\(628\) −4.45210e7 −0.00717310
\(629\) 6.24962e8 0.100133
\(630\) −7.24210e8 −0.115391
\(631\) −3.40983e9 −0.540294 −0.270147 0.962819i \(-0.587072\pi\)
−0.270147 + 0.962819i \(0.587072\pi\)
\(632\) −8.98098e9 −1.41519
\(633\) 5.52204e9 0.865339
\(634\) 1.25096e10 1.94954
\(635\) 6.95733e9 1.07829
\(636\) 5.17247e7 0.00797257
\(637\) 9.84077e9 1.50849
\(638\) 1.21084e10 1.84593
\(639\) −3.42856e9 −0.519827
\(640\) −1.01751e10 −1.53430
\(641\) −7.05809e9 −1.05848 −0.529242 0.848471i \(-0.677524\pi\)
−0.529242 + 0.848471i \(0.677524\pi\)
\(642\) 6.14119e9 0.915967
\(643\) 9.53642e9 1.41464 0.707322 0.706892i \(-0.249903\pi\)
0.707322 + 0.706892i \(0.249903\pi\)
\(644\) 7.28220e7 0.0107439
\(645\) 7.80005e9 1.14456
\(646\) −1.33069e10 −1.94206
\(647\) 1.17077e10 1.69945 0.849723 0.527230i \(-0.176769\pi\)
0.849723 + 0.527230i \(0.176769\pi\)
\(648\) −7.48903e8 −0.108122
\(649\) 1.02158e9 0.146696
\(650\) −1.21406e10 −1.73398
\(651\) 6.84202e7 0.00971965
\(652\) −1.23066e8 −0.0173889
\(653\) −1.32050e9 −0.185584 −0.0927921 0.995686i \(-0.529579\pi\)
−0.0927921 + 0.995686i \(0.529579\pi\)
\(654\) −7.99784e9 −1.11802
\(655\) 5.46499e9 0.759879
\(656\) −1.37553e10 −1.90242
\(657\) −2.66505e9 −0.366629
\(658\) −1.34742e8 −0.0184379
\(659\) −2.98482e9 −0.406275 −0.203137 0.979150i \(-0.565114\pi\)
−0.203137 + 0.979150i \(0.565114\pi\)
\(660\) −3.49236e8 −0.0472841
\(661\) −1.22973e10 −1.65617 −0.828087 0.560600i \(-0.810571\pi\)
−0.828087 + 0.560600i \(0.810571\pi\)
\(662\) 2.22464e9 0.298027
\(663\) 1.26158e10 1.68119
\(664\) 2.94100e9 0.389858
\(665\) 2.66124e9 0.350920
\(666\) 1.43038e8 0.0187626
\(667\) −1.10401e10 −1.44056
\(668\) −1.10460e8 −0.0143380
\(669\) −6.58968e9 −0.850889
\(670\) −1.41966e10 −1.82358
\(671\) −6.81592e9 −0.870955
\(672\) 1.08174e8 0.0137509
\(673\) −7.89633e9 −0.998557 −0.499278 0.866442i \(-0.666401\pi\)
−0.499278 + 0.866442i \(0.666401\pi\)
\(674\) 5.48584e9 0.690134
\(675\) −1.62896e9 −0.203867
\(676\) 6.30625e8 0.0785159
\(677\) −1.34037e9 −0.166021 −0.0830106 0.996549i \(-0.526454\pi\)
−0.0830106 + 0.996549i \(0.526454\pi\)
\(678\) 3.90716e9 0.481457
\(679\) −1.37262e9 −0.168270
\(680\) 2.08780e10 2.54629
\(681\) 4.79020e9 0.581218
\(682\) 6.84425e8 0.0826191
\(683\) −6.85009e9 −0.822667 −0.411333 0.911485i \(-0.634937\pi\)
−0.411333 + 0.911485i \(0.634937\pi\)
\(684\) −1.46821e8 −0.0175425
\(685\) 6.71080e9 0.797732
\(686\) −3.96643e9 −0.469100
\(687\) −5.87465e9 −0.691247
\(688\) −1.23678e10 −1.44788
\(689\) 3.73803e9 0.435387
\(690\) 6.60528e9 0.765454
\(691\) 1.48604e10 1.71340 0.856698 0.515818i \(-0.172512\pi\)
0.856698 + 0.515818i \(0.172512\pi\)
\(692\) −5.23599e7 −0.00600658
\(693\) −7.74447e8 −0.0883945
\(694\) −2.77662e9 −0.315325
\(695\) −4.69668e9 −0.530694
\(696\) −7.98677e9 −0.897922
\(697\) 2.95880e10 3.30980
\(698\) 2.96055e9 0.329518
\(699\) 8.85097e9 0.980214
\(700\) 1.14589e8 0.0126270
\(701\) −1.54673e10 −1.69590 −0.847952 0.530073i \(-0.822164\pi\)
−0.847952 + 0.530073i \(0.822164\pi\)
\(702\) 2.88745e9 0.315017
\(703\) −5.25620e8 −0.0570596
\(704\) −9.85102e9 −1.06409
\(705\) −5.89174e8 −0.0633260
\(706\) 1.06319e10 1.13709
\(707\) 2.27934e8 0.0242572
\(708\) 3.59501e7 0.00380701
\(709\) 2.09686e9 0.220957 0.110478 0.993879i \(-0.464762\pi\)
0.110478 + 0.993879i \(0.464762\pi\)
\(710\) 2.18764e10 2.29388
\(711\) 4.64601e9 0.484771
\(712\) −2.02422e9 −0.210173
\(713\) −6.24037e8 −0.0644759
\(714\) −2.47005e9 −0.253958
\(715\) −2.52385e10 −2.58222
\(716\) 3.24119e8 0.0329996
\(717\) −7.54547e9 −0.764485
\(718\) −1.23065e9 −0.124079
\(719\) −1.96050e10 −1.96705 −0.983527 0.180761i \(-0.942144\pi\)
−0.983527 + 0.180761i \(0.942144\pi\)
\(720\) 5.02112e9 0.501345
\(721\) −2.09431e9 −0.208097
\(722\) 8.25721e8 0.0816494
\(723\) −1.03486e10 −1.01835
\(724\) 4.58622e8 0.0449128
\(725\) −1.73722e10 −1.69306
\(726\) −1.64536e9 −0.159582
\(727\) −1.11617e10 −1.07735 −0.538677 0.842513i \(-0.681075\pi\)
−0.538677 + 0.842513i \(0.681075\pi\)
\(728\) 3.80720e9 0.365717
\(729\) 3.87420e8 0.0370370
\(730\) 1.70047e10 1.61785
\(731\) 2.66035e10 2.51899
\(732\) −2.39856e8 −0.0226028
\(733\) 1.37783e10 1.29221 0.646103 0.763250i \(-0.276397\pi\)
0.646103 + 0.763250i \(0.276397\pi\)
\(734\) 1.31437e10 1.22683
\(735\) −8.42483e9 −0.782628
\(736\) −9.86621e8 −0.0912175
\(737\) −1.51814e10 −1.39694
\(738\) 6.77198e9 0.620181
\(739\) 3.03220e9 0.276377 0.138189 0.990406i \(-0.455872\pi\)
0.138189 + 0.990406i \(0.455872\pi\)
\(740\) −4.39976e7 −0.00399133
\(741\) −1.06104e10 −0.958008
\(742\) −7.31870e8 −0.0657689
\(743\) −1.93427e9 −0.173004 −0.0865020 0.996252i \(-0.527569\pi\)
−0.0865020 + 0.996252i \(0.527569\pi\)
\(744\) −4.51449e8 −0.0401886
\(745\) −6.15009e7 −0.00544923
\(746\) 4.04318e9 0.356563
\(747\) −1.52143e9 −0.133545
\(748\) −1.19113e9 −0.104065
\(749\) −4.18891e9 −0.364263
\(750\) 5.82060e8 0.0503795
\(751\) 5.54431e9 0.477647 0.238824 0.971063i \(-0.423238\pi\)
0.238824 + 0.971063i \(0.423238\pi\)
\(752\) 9.34197e8 0.0801081
\(753\) 1.14466e10 0.977000
\(754\) 3.07935e10 2.61613
\(755\) −3.06761e10 −2.59409
\(756\) −2.72532e7 −0.00229399
\(757\) 1.76557e10 1.47927 0.739637 0.673006i \(-0.234997\pi\)
0.739637 + 0.673006i \(0.234997\pi\)
\(758\) −2.29332e9 −0.191259
\(759\) 7.06347e9 0.586370
\(760\) −1.75593e10 −1.45098
\(761\) 2.32983e9 0.191636 0.0958180 0.995399i \(-0.469453\pi\)
0.0958180 + 0.995399i \(0.469453\pi\)
\(762\) 5.43104e9 0.444673
\(763\) 5.45534e9 0.444617
\(764\) −9.09095e8 −0.0737535
\(765\) −1.08006e10 −0.872231
\(766\) −1.85918e10 −1.49458
\(767\) 2.59803e9 0.207903
\(768\) −1.09850e9 −0.0875052
\(769\) 8.88822e9 0.704811 0.352405 0.935847i \(-0.385364\pi\)
0.352405 + 0.935847i \(0.385364\pi\)
\(770\) 4.94146e9 0.390066
\(771\) −4.88332e9 −0.383729
\(772\) −3.77138e8 −0.0295012
\(773\) −1.51855e9 −0.118250 −0.0591251 0.998251i \(-0.518831\pi\)
−0.0591251 + 0.998251i \(0.518831\pi\)
\(774\) 6.08888e9 0.472002
\(775\) −9.81957e8 −0.0757769
\(776\) 9.05679e9 0.695758
\(777\) −9.75667e7 −0.00746153
\(778\) −8.98038e9 −0.683701
\(779\) −2.48848e10 −1.88605
\(780\) −8.88157e8 −0.0670130
\(781\) 2.33939e10 1.75721
\(782\) 2.25285e10 1.68464
\(783\) 4.13169e9 0.307583
\(784\) 1.33585e10 0.990034
\(785\) 2.75450e9 0.203235
\(786\) 4.26608e9 0.313365
\(787\) 5.14294e9 0.376097 0.188048 0.982160i \(-0.439784\pi\)
0.188048 + 0.982160i \(0.439784\pi\)
\(788\) 3.92984e8 0.0286110
\(789\) −1.14397e10 −0.829170
\(790\) −2.96445e10 −2.13919
\(791\) −2.66508e9 −0.191466
\(792\) 5.10994e9 0.365492
\(793\) −1.73339e10 −1.23435
\(794\) −1.44879e10 −1.02715
\(795\) −3.20019e9 −0.225886
\(796\) 1.51925e6 0.000106766 0
\(797\) 2.31164e10 1.61739 0.808697 0.588225i \(-0.200173\pi\)
0.808697 + 0.588225i \(0.200173\pi\)
\(798\) 2.07742e9 0.144715
\(799\) −2.00948e9 −0.139371
\(800\) −1.55250e9 −0.107206
\(801\) 1.04716e9 0.0719947
\(802\) −2.43580e10 −1.66737
\(803\) 1.81843e10 1.23934
\(804\) −5.34243e8 −0.0362529
\(805\) −4.50547e9 −0.304407
\(806\) 1.74059e9 0.117091
\(807\) −9.40729e9 −0.630096
\(808\) −1.50395e9 −0.100298
\(809\) 2.29952e10 1.52692 0.763462 0.645853i \(-0.223498\pi\)
0.763462 + 0.645853i \(0.223498\pi\)
\(810\) −2.47199e9 −0.163436
\(811\) −2.76002e9 −0.181694 −0.0908468 0.995865i \(-0.528957\pi\)
−0.0908468 + 0.995865i \(0.528957\pi\)
\(812\) −2.90645e8 −0.0190510
\(813\) 3.41391e9 0.222810
\(814\) −9.75985e8 −0.0634246
\(815\) 7.61407e9 0.492680
\(816\) 1.71254e10 1.10338
\(817\) −2.23747e10 −1.43542
\(818\) 2.43153e10 1.55326
\(819\) −1.96953e9 −0.125276
\(820\) −2.08301e9 −0.131930
\(821\) 5.67826e9 0.358108 0.179054 0.983839i \(-0.442696\pi\)
0.179054 + 0.983839i \(0.442696\pi\)
\(822\) 5.23859e9 0.328975
\(823\) −2.56662e7 −0.00160495 −0.000802475 1.00000i \(-0.500255\pi\)
−0.000802475 1.00000i \(0.500255\pi\)
\(824\) 1.38186e10 0.860437
\(825\) 1.11148e10 0.689146
\(826\) −5.08670e8 −0.0314055
\(827\) −2.08817e10 −1.28380 −0.641899 0.766789i \(-0.721853\pi\)
−0.641899 + 0.766789i \(0.721853\pi\)
\(828\) 2.48567e8 0.0152173
\(829\) −2.54198e10 −1.54964 −0.774821 0.632180i \(-0.782160\pi\)
−0.774821 + 0.632180i \(0.782160\pi\)
\(830\) 9.70767e9 0.589307
\(831\) 5.99654e9 0.362491
\(832\) −2.50525e10 −1.50807
\(833\) −2.87344e10 −1.72244
\(834\) −3.66633e9 −0.218852
\(835\) 6.83412e9 0.406237
\(836\) 1.00179e9 0.0593003
\(837\) 2.33542e8 0.0137666
\(838\) −8.74999e9 −0.513634
\(839\) −3.56300e9 −0.208281 −0.104140 0.994563i \(-0.533209\pi\)
−0.104140 + 0.994563i \(0.533209\pi\)
\(840\) −3.25940e9 −0.189741
\(841\) 2.68130e10 1.55439
\(842\) −2.12325e9 −0.122577
\(843\) 8.64776e9 0.497172
\(844\) −1.32592e9 −0.0759132
\(845\) −3.90165e10 −2.22459
\(846\) −4.59922e8 −0.0261149
\(847\) 1.12230e9 0.0634627
\(848\) 5.07423e9 0.285749
\(849\) −6.64908e9 −0.372893
\(850\) 3.54498e10 1.97992
\(851\) 8.89873e8 0.0494965
\(852\) 8.23243e8 0.0456026
\(853\) 2.29229e10 1.26458 0.632291 0.774731i \(-0.282115\pi\)
0.632291 + 0.774731i \(0.282115\pi\)
\(854\) 3.39380e9 0.186459
\(855\) 9.08375e9 0.497031
\(856\) 2.76392e10 1.50615
\(857\) −1.48684e10 −0.806924 −0.403462 0.914996i \(-0.632193\pi\)
−0.403462 + 0.914996i \(0.632193\pi\)
\(858\) −1.97017e10 −1.06488
\(859\) −2.17432e10 −1.17043 −0.585217 0.810877i \(-0.698991\pi\)
−0.585217 + 0.810877i \(0.698991\pi\)
\(860\) −1.87290e9 −0.100408
\(861\) −4.61917e9 −0.246634
\(862\) 3.16004e9 0.168042
\(863\) −1.50753e10 −0.798414 −0.399207 0.916861i \(-0.630715\pi\)
−0.399207 + 0.916861i \(0.630715\pi\)
\(864\) 3.69237e8 0.0194763
\(865\) 3.23948e9 0.170184
\(866\) −1.05169e10 −0.550269
\(867\) −2.57581e10 −1.34229
\(868\) −1.64286e7 −0.000852671 0
\(869\) −3.17009e10 −1.63871
\(870\) −2.63628e10 −1.35729
\(871\) −3.86085e10 −1.97979
\(872\) −3.59953e10 −1.83839
\(873\) −4.68523e9 −0.238331
\(874\) −1.89474e10 −0.959977
\(875\) −3.97024e8 −0.0200350
\(876\) 6.39914e8 0.0321630
\(877\) −1.29162e10 −0.646598 −0.323299 0.946297i \(-0.604792\pi\)
−0.323299 + 0.946297i \(0.604792\pi\)
\(878\) −7.58858e8 −0.0378381
\(879\) −6.10213e9 −0.303054
\(880\) −3.42603e10 −1.69473
\(881\) −2.47400e10 −1.21895 −0.609473 0.792807i \(-0.708619\pi\)
−0.609473 + 0.792807i \(0.708619\pi\)
\(882\) −6.57660e9 −0.322747
\(883\) 2.21032e10 1.08042 0.540211 0.841530i \(-0.318345\pi\)
0.540211 + 0.841530i \(0.318345\pi\)
\(884\) −3.02922e9 −0.147485
\(885\) −2.22422e9 −0.107864
\(886\) 3.01569e10 1.45670
\(887\) −8.79294e9 −0.423060 −0.211530 0.977372i \(-0.567845\pi\)
−0.211530 + 0.977372i \(0.567845\pi\)
\(888\) 6.43763e8 0.0308518
\(889\) −3.70452e9 −0.176838
\(890\) −6.68156e9 −0.317697
\(891\) −2.64346e9 −0.125199
\(892\) 1.58227e9 0.0746455
\(893\) 1.69006e9 0.0794188
\(894\) −4.80089e7 −0.00224719
\(895\) −2.00531e10 −0.934976
\(896\) 5.41787e9 0.251623
\(897\) 1.79634e10 0.831027
\(898\) 2.93239e10 1.35131
\(899\) 2.49064e9 0.114328
\(900\) 3.91134e8 0.0178845
\(901\) −1.09148e10 −0.497141
\(902\) −4.62068e10 −2.09644
\(903\) −4.15323e9 −0.187706
\(904\) 1.75847e10 0.791671
\(905\) −2.83747e10 −1.27251
\(906\) −2.39464e10 −1.06977
\(907\) −6.78075e8 −0.0301753 −0.0150877 0.999886i \(-0.504803\pi\)
−0.0150877 + 0.999886i \(0.504803\pi\)
\(908\) −1.15019e9 −0.0509882
\(909\) 7.78019e8 0.0343571
\(910\) 1.25668e10 0.552817
\(911\) 1.87905e10 0.823424 0.411712 0.911314i \(-0.364931\pi\)
0.411712 + 0.911314i \(0.364931\pi\)
\(912\) −1.44032e10 −0.628750
\(913\) 1.03811e10 0.451434
\(914\) −1.47148e10 −0.637443
\(915\) 1.48398e10 0.640404
\(916\) 1.41058e9 0.0606407
\(917\) −2.90990e9 −0.124619
\(918\) −8.43115e9 −0.359698
\(919\) 3.67497e10 1.56189 0.780943 0.624603i \(-0.214739\pi\)
0.780943 + 0.624603i \(0.214739\pi\)
\(920\) 2.97279e10 1.25866
\(921\) −1.49497e10 −0.630557
\(922\) 7.16352e9 0.301001
\(923\) 5.94939e10 2.49039
\(924\) 1.85955e8 0.00775454
\(925\) 1.40026e9 0.0581720
\(926\) 2.05128e10 0.848957
\(927\) −7.14861e9 −0.294742
\(928\) 3.93777e9 0.161746
\(929\) 5.36643e9 0.219599 0.109800 0.993954i \(-0.464979\pi\)
0.109800 + 0.993954i \(0.464979\pi\)
\(930\) −1.49015e9 −0.0607489
\(931\) 2.41669e10 0.981515
\(932\) −2.12524e9 −0.0859907
\(933\) −2.35456e10 −0.949127
\(934\) 2.42599e10 0.974262
\(935\) 7.36948e10 2.94847
\(936\) 1.29953e10 0.517990
\(937\) −2.19515e10 −0.871717 −0.435859 0.900015i \(-0.643555\pi\)
−0.435859 + 0.900015i \(0.643555\pi\)
\(938\) 7.55918e9 0.299064
\(939\) −8.84869e9 −0.348778
\(940\) 1.41469e8 0.00555537
\(941\) 2.82488e10 1.10519 0.552595 0.833450i \(-0.313638\pi\)
0.552595 + 0.833450i \(0.313638\pi\)
\(942\) 2.15022e9 0.0838117
\(943\) 4.21299e10 1.63606
\(944\) 3.52673e9 0.136449
\(945\) 1.68614e9 0.0649955
\(946\) −4.15459e10 −1.59554
\(947\) 2.51489e10 0.962264 0.481132 0.876648i \(-0.340226\pi\)
0.481132 + 0.876648i \(0.340226\pi\)
\(948\) −1.11557e9 −0.0425273
\(949\) 4.62452e10 1.75645
\(950\) −2.98149e10 −1.12824
\(951\) −2.91255e10 −1.09810
\(952\) −1.11168e10 −0.417589
\(953\) −4.09484e10 −1.53254 −0.766269 0.642519i \(-0.777889\pi\)
−0.766269 + 0.642519i \(0.777889\pi\)
\(954\) −2.49813e9 −0.0931529
\(955\) 5.62453e10 2.08966
\(956\) 1.81177e9 0.0670656
\(957\) −2.81915e10 −1.03974
\(958\) −8.37249e9 −0.307663
\(959\) −3.57325e9 −0.130827
\(960\) 2.14479e10 0.782411
\(961\) −2.73718e10 −0.994883
\(962\) −2.48207e9 −0.0898879
\(963\) −1.42982e10 −0.515930
\(964\) 2.48483e9 0.0893360
\(965\) 2.33334e10 0.835858
\(966\) −3.51706e9 −0.125534
\(967\) −5.25463e10 −1.86874 −0.934372 0.356299i \(-0.884038\pi\)
−0.934372 + 0.356299i \(0.884038\pi\)
\(968\) −7.40517e9 −0.262404
\(969\) 3.09818e10 1.09389
\(970\) 2.98947e10 1.05170
\(971\) −2.49973e9 −0.0876246 −0.0438123 0.999040i \(-0.513950\pi\)
−0.0438123 + 0.999040i \(0.513950\pi\)
\(972\) −9.30248e7 −0.00324913
\(973\) 2.50081e9 0.0870332
\(974\) 2.93482e10 1.01771
\(975\) 2.82664e10 0.976686
\(976\) −2.35300e10 −0.810118
\(977\) −2.71796e10 −0.932420 −0.466210 0.884674i \(-0.654381\pi\)
−0.466210 + 0.884674i \(0.654381\pi\)
\(978\) 5.94370e9 0.203175
\(979\) −7.14505e9 −0.243369
\(980\) 2.02292e9 0.0686573
\(981\) 1.86210e10 0.629740
\(982\) 4.66316e10 1.57141
\(983\) 3.97905e9 0.133611 0.0668055 0.997766i \(-0.478719\pi\)
0.0668055 + 0.997766i \(0.478719\pi\)
\(984\) 3.04781e10 1.01978
\(985\) −2.43138e10 −0.810635
\(986\) −8.99150e10 −2.98719
\(987\) 3.13713e8 0.0103854
\(988\) 2.54771e9 0.0840427
\(989\) 3.78802e10 1.24516
\(990\) 1.68670e10 0.552476
\(991\) −2.12813e9 −0.0694610 −0.0347305 0.999397i \(-0.511057\pi\)
−0.0347305 + 0.999397i \(0.511057\pi\)
\(992\) 2.22581e8 0.00723931
\(993\) −5.17951e9 −0.167867
\(994\) −1.16483e10 −0.376194
\(995\) −9.39956e7 −0.00302501
\(996\) 3.65315e8 0.0117155
\(997\) −1.59834e8 −0.00510784 −0.00255392 0.999997i \(-0.500813\pi\)
−0.00255392 + 0.999997i \(0.500813\pi\)
\(998\) −1.40932e10 −0.448800
\(999\) −3.33030e8 −0.0105683
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.8.a.c.1.13 17
3.2 odd 2 531.8.a.c.1.5 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.13 17 1.1 even 1 trivial
531.8.a.c.1.5 17 3.2 odd 2