Properties

Label 177.8.a.c.1.2
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 1.2
Root \(-20.9857\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-20.9857 q^{2} -27.0000 q^{3} +312.401 q^{4} -465.631 q^{5} +566.615 q^{6} +286.579 q^{7} -3869.79 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-20.9857 q^{2} -27.0000 q^{3} +312.401 q^{4} -465.631 q^{5} +566.615 q^{6} +286.579 q^{7} -3869.79 q^{8} +729.000 q^{9} +9771.60 q^{10} -3497.66 q^{11} -8434.83 q^{12} -4677.13 q^{13} -6014.07 q^{14} +12572.0 q^{15} +41223.0 q^{16} -1899.70 q^{17} -15298.6 q^{18} +6396.17 q^{19} -145463. q^{20} -7737.63 q^{21} +73400.9 q^{22} +9375.09 q^{23} +104484. q^{24} +138687. q^{25} +98153.0 q^{26} -19683.0 q^{27} +89527.5 q^{28} -160070. q^{29} -263833. q^{30} -272299. q^{31} -369763. q^{32} +94436.8 q^{33} +39866.7 q^{34} -133440. q^{35} +227740. q^{36} -35341.8 q^{37} -134228. q^{38} +126283. q^{39} +1.80189e6 q^{40} -727812. q^{41} +162380. q^{42} -712974. q^{43} -1.09267e6 q^{44} -339445. q^{45} -196743. q^{46} -476612. q^{47} -1.11302e6 q^{48} -741416. q^{49} -2.91045e6 q^{50} +51292.0 q^{51} -1.46114e6 q^{52} -1.49764e6 q^{53} +413062. q^{54} +1.62862e6 q^{55} -1.10900e6 q^{56} -172697. q^{57} +3.35919e6 q^{58} -205379. q^{59} +3.92751e6 q^{60} -1.04757e6 q^{61} +5.71439e6 q^{62} +208916. q^{63} +2.48319e6 q^{64} +2.17782e6 q^{65} -1.98182e6 q^{66} -2.56238e6 q^{67} -593470. q^{68} -253127. q^{69} +2.80033e6 q^{70} +3.16847e6 q^{71} -2.82108e6 q^{72} +1.62960e6 q^{73} +741674. q^{74} -3.74455e6 q^{75} +1.99817e6 q^{76} -1.00235e6 q^{77} -2.65013e6 q^{78} -2.11177e6 q^{79} -1.91947e7 q^{80} +531441. q^{81} +1.52737e7 q^{82} +5.43782e6 q^{83} -2.41724e6 q^{84} +884561. q^{85} +1.49623e7 q^{86} +4.32189e6 q^{87} +1.35352e7 q^{88} -2.27504e6 q^{89} +7.12350e6 q^{90} -1.34037e6 q^{91} +2.92879e6 q^{92} +7.35206e6 q^{93} +1.00021e7 q^{94} -2.97825e6 q^{95} +9.98359e6 q^{96} +1.00026e7 q^{97} +1.55591e7 q^{98} -2.54979e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −20.9857 −1.85489 −0.927447 0.373954i \(-0.878002\pi\)
−0.927447 + 0.373954i \(0.878002\pi\)
\(3\) −27.0000 −0.577350
\(4\) 312.401 2.44063
\(5\) −465.631 −1.66589 −0.832945 0.553355i \(-0.813347\pi\)
−0.832945 + 0.553355i \(0.813347\pi\)
\(6\) 566.615 1.07092
\(7\) 286.579 0.315792 0.157896 0.987456i \(-0.449529\pi\)
0.157896 + 0.987456i \(0.449529\pi\)
\(8\) −3869.79 −2.67222
\(9\) 729.000 0.333333
\(10\) 9771.60 3.09005
\(11\) −3497.66 −0.792324 −0.396162 0.918181i \(-0.629658\pi\)
−0.396162 + 0.918181i \(0.629658\pi\)
\(12\) −8434.83 −1.40910
\(13\) −4677.13 −0.590443 −0.295221 0.955429i \(-0.595393\pi\)
−0.295221 + 0.955429i \(0.595393\pi\)
\(14\) −6014.07 −0.585761
\(15\) 12572.0 0.961802
\(16\) 41223.0 2.51605
\(17\) −1899.70 −0.0937810 −0.0468905 0.998900i \(-0.514931\pi\)
−0.0468905 + 0.998900i \(0.514931\pi\)
\(18\) −15298.6 −0.618298
\(19\) 6396.17 0.213935 0.106968 0.994263i \(-0.465886\pi\)
0.106968 + 0.994263i \(0.465886\pi\)
\(20\) −145463. −4.06583
\(21\) −7737.63 −0.182323
\(22\) 73400.9 1.46968
\(23\) 9375.09 0.160668 0.0803338 0.996768i \(-0.474401\pi\)
0.0803338 + 0.996768i \(0.474401\pi\)
\(24\) 104484. 1.54281
\(25\) 138687. 1.77519
\(26\) 98153.0 1.09521
\(27\) −19683.0 −0.192450
\(28\) 89527.5 0.770732
\(29\) −160070. −1.21876 −0.609379 0.792879i \(-0.708581\pi\)
−0.609379 + 0.792879i \(0.708581\pi\)
\(30\) −263833. −1.78404
\(31\) −272299. −1.64165 −0.820823 0.571183i \(-0.806485\pi\)
−0.820823 + 0.571183i \(0.806485\pi\)
\(32\) −369763. −1.99479
\(33\) 94436.8 0.457449
\(34\) 39866.7 0.173954
\(35\) −133440. −0.526075
\(36\) 227740. 0.813544
\(37\) −35341.8 −0.114705 −0.0573525 0.998354i \(-0.518266\pi\)
−0.0573525 + 0.998354i \(0.518266\pi\)
\(38\) −134228. −0.396827
\(39\) 126283. 0.340892
\(40\) 1.80189e6 4.45163
\(41\) −727812. −1.64921 −0.824605 0.565709i \(-0.808603\pi\)
−0.824605 + 0.565709i \(0.808603\pi\)
\(42\) 162380. 0.338189
\(43\) −712974. −1.36752 −0.683761 0.729706i \(-0.739657\pi\)
−0.683761 + 0.729706i \(0.739657\pi\)
\(44\) −1.09267e6 −1.93377
\(45\) −339445. −0.555297
\(46\) −196743. −0.298021
\(47\) −476612. −0.669611 −0.334805 0.942287i \(-0.608671\pi\)
−0.334805 + 0.942287i \(0.608671\pi\)
\(48\) −1.11302e6 −1.45264
\(49\) −741416. −0.900275
\(50\) −2.91045e6 −3.29279
\(51\) 51292.0 0.0541445
\(52\) −1.46114e6 −1.44105
\(53\) −1.49764e6 −1.38179 −0.690894 0.722956i \(-0.742783\pi\)
−0.690894 + 0.722956i \(0.742783\pi\)
\(54\) 413062. 0.356975
\(55\) 1.62862e6 1.31993
\(56\) −1.10900e6 −0.843866
\(57\) −172697. −0.123516
\(58\) 3.35919e6 2.26067
\(59\) −205379. −0.130189
\(60\) 3.92751e6 2.34741
\(61\) −1.04757e6 −0.590919 −0.295460 0.955355i \(-0.595473\pi\)
−0.295460 + 0.955355i \(0.595473\pi\)
\(62\) 5.71439e6 3.04508
\(63\) 208916. 0.105264
\(64\) 2.48319e6 1.18408
\(65\) 2.17782e6 0.983613
\(66\) −1.98182e6 −0.848519
\(67\) −2.56238e6 −1.04083 −0.520416 0.853913i \(-0.674223\pi\)
−0.520416 + 0.853913i \(0.674223\pi\)
\(68\) −593470. −0.228885
\(69\) −253127. −0.0927614
\(70\) 2.80033e6 0.975813
\(71\) 3.16847e6 1.05062 0.525309 0.850912i \(-0.323950\pi\)
0.525309 + 0.850912i \(0.323950\pi\)
\(72\) −2.82108e6 −0.890740
\(73\) 1.62960e6 0.490287 0.245144 0.969487i \(-0.421165\pi\)
0.245144 + 0.969487i \(0.421165\pi\)
\(74\) 741674. 0.212766
\(75\) −3.74455e6 −1.02491
\(76\) 1.99817e6 0.522138
\(77\) −1.00235e6 −0.250210
\(78\) −2.65013e6 −0.632319
\(79\) −2.11177e6 −0.481893 −0.240947 0.970538i \(-0.577458\pi\)
−0.240947 + 0.970538i \(0.577458\pi\)
\(80\) −1.91947e7 −4.19147
\(81\) 531441. 0.111111
\(82\) 1.52737e7 3.05911
\(83\) 5.43782e6 1.04388 0.521941 0.852982i \(-0.325208\pi\)
0.521941 + 0.852982i \(0.325208\pi\)
\(84\) −2.41724e6 −0.444982
\(85\) 884561. 0.156229
\(86\) 1.49623e7 2.53661
\(87\) 4.32189e6 0.703650
\(88\) 1.35352e7 2.11727
\(89\) −2.27504e6 −0.342078 −0.171039 0.985264i \(-0.554712\pi\)
−0.171039 + 0.985264i \(0.554712\pi\)
\(90\) 7.12350e6 1.03002
\(91\) −1.34037e6 −0.186457
\(92\) 2.92879e6 0.392130
\(93\) 7.35206e6 0.947805
\(94\) 1.00021e7 1.24206
\(95\) −2.97825e6 −0.356393
\(96\) 9.98359e6 1.15170
\(97\) 1.00026e7 1.11279 0.556395 0.830918i \(-0.312184\pi\)
0.556395 + 0.830918i \(0.312184\pi\)
\(98\) 1.55591e7 1.66992
\(99\) −2.54979e6 −0.264108
\(100\) 4.33259e7 4.33259
\(101\) −4.08392e6 −0.394414 −0.197207 0.980362i \(-0.563187\pi\)
−0.197207 + 0.980362i \(0.563187\pi\)
\(102\) −1.07640e6 −0.100432
\(103\) −1.63687e7 −1.47599 −0.737996 0.674805i \(-0.764228\pi\)
−0.737996 + 0.674805i \(0.764228\pi\)
\(104\) 1.80995e7 1.57779
\(105\) 3.60288e6 0.303729
\(106\) 3.14290e7 2.56307
\(107\) −1.43241e7 −1.13038 −0.565189 0.824961i \(-0.691197\pi\)
−0.565189 + 0.824961i \(0.691197\pi\)
\(108\) −6.14899e6 −0.469700
\(109\) −666192. −0.0492728 −0.0246364 0.999696i \(-0.507843\pi\)
−0.0246364 + 0.999696i \(0.507843\pi\)
\(110\) −3.41777e7 −2.44832
\(111\) 954229. 0.0662250
\(112\) 1.18137e7 0.794550
\(113\) 2.37986e7 1.55159 0.775795 0.630986i \(-0.217349\pi\)
0.775795 + 0.630986i \(0.217349\pi\)
\(114\) 3.62417e6 0.229108
\(115\) −4.36533e6 −0.267655
\(116\) −5.00060e7 −2.97454
\(117\) −3.40963e6 −0.196814
\(118\) 4.31003e6 0.241487
\(119\) −544415. −0.0296153
\(120\) −4.86511e7 −2.57015
\(121\) −7.25356e6 −0.372222
\(122\) 2.19840e7 1.09609
\(123\) 1.96509e7 0.952172
\(124\) −8.50663e7 −4.00665
\(125\) −2.81995e7 −1.29139
\(126\) −4.38426e6 −0.195254
\(127\) 7.78497e6 0.337244 0.168622 0.985681i \(-0.446068\pi\)
0.168622 + 0.985681i \(0.446068\pi\)
\(128\) −4.78197e6 −0.201545
\(129\) 1.92503e7 0.789539
\(130\) −4.57031e7 −1.82450
\(131\) −2.68679e7 −1.04420 −0.522101 0.852884i \(-0.674852\pi\)
−0.522101 + 0.852884i \(0.674852\pi\)
\(132\) 2.95021e7 1.11646
\(133\) 1.83301e6 0.0675590
\(134\) 5.37733e7 1.93063
\(135\) 9.16501e6 0.320601
\(136\) 7.35146e6 0.250604
\(137\) −5.02442e7 −1.66941 −0.834707 0.550695i \(-0.814363\pi\)
−0.834707 + 0.550695i \(0.814363\pi\)
\(138\) 5.31207e6 0.172063
\(139\) −5.46930e7 −1.72735 −0.863675 0.504049i \(-0.831843\pi\)
−0.863675 + 0.504049i \(0.831843\pi\)
\(140\) −4.16868e7 −1.28396
\(141\) 1.28685e7 0.386600
\(142\) −6.64926e7 −1.94879
\(143\) 1.63590e7 0.467822
\(144\) 3.00516e7 0.838685
\(145\) 7.45335e7 2.03032
\(146\) −3.41983e7 −0.909431
\(147\) 2.00182e7 0.519774
\(148\) −1.10408e7 −0.279953
\(149\) 3.71390e7 0.919769 0.459884 0.887979i \(-0.347891\pi\)
0.459884 + 0.887979i \(0.347891\pi\)
\(150\) 7.85820e7 1.90110
\(151\) 1.36501e7 0.322639 0.161319 0.986902i \(-0.448425\pi\)
0.161319 + 0.986902i \(0.448425\pi\)
\(152\) −2.47518e7 −0.571682
\(153\) −1.38488e6 −0.0312603
\(154\) 2.10352e7 0.464112
\(155\) 1.26791e8 2.73480
\(156\) 3.94508e7 0.831993
\(157\) 3.77615e7 0.778755 0.389378 0.921078i \(-0.372690\pi\)
0.389378 + 0.921078i \(0.372690\pi\)
\(158\) 4.43169e7 0.893861
\(159\) 4.04362e7 0.797776
\(160\) 1.72173e8 3.32311
\(161\) 2.68670e6 0.0507375
\(162\) −1.11527e7 −0.206099
\(163\) 4.92529e7 0.890790 0.445395 0.895334i \(-0.353063\pi\)
0.445395 + 0.895334i \(0.353063\pi\)
\(164\) −2.27369e8 −4.02512
\(165\) −4.39726e7 −0.762059
\(166\) −1.14117e8 −1.93629
\(167\) 1.04186e8 1.73102 0.865508 0.500895i \(-0.166996\pi\)
0.865508 + 0.500895i \(0.166996\pi\)
\(168\) 2.99430e7 0.487206
\(169\) −4.08730e7 −0.651377
\(170\) −1.85632e7 −0.289788
\(171\) 4.66281e6 0.0713118
\(172\) −2.22734e8 −3.33762
\(173\) 9.60679e7 1.41064 0.705321 0.708888i \(-0.250803\pi\)
0.705321 + 0.708888i \(0.250803\pi\)
\(174\) −9.06981e7 −1.30520
\(175\) 3.97447e7 0.560591
\(176\) −1.44184e8 −1.99353
\(177\) 5.54523e6 0.0751646
\(178\) 4.77435e7 0.634518
\(179\) −8.54665e7 −1.11381 −0.556904 0.830577i \(-0.688011\pi\)
−0.556904 + 0.830577i \(0.688011\pi\)
\(180\) −1.06043e8 −1.35528
\(181\) 4.14050e7 0.519012 0.259506 0.965741i \(-0.416440\pi\)
0.259506 + 0.965741i \(0.416440\pi\)
\(182\) 2.81286e7 0.345858
\(183\) 2.82844e7 0.341167
\(184\) −3.62796e7 −0.429339
\(185\) 1.64562e7 0.191086
\(186\) −1.54288e8 −1.75808
\(187\) 6.64452e6 0.0743050
\(188\) −1.48894e8 −1.63427
\(189\) −5.64073e6 −0.0607742
\(190\) 6.25008e7 0.661071
\(191\) −8.00346e7 −0.831114 −0.415557 0.909567i \(-0.636413\pi\)
−0.415557 + 0.909567i \(0.636413\pi\)
\(192\) −6.70462e7 −0.683628
\(193\) 1.36187e8 1.36360 0.681798 0.731540i \(-0.261198\pi\)
0.681798 + 0.731540i \(0.261198\pi\)
\(194\) −2.09913e8 −2.06411
\(195\) −5.88010e7 −0.567889
\(196\) −2.31619e8 −2.19724
\(197\) −1.36747e8 −1.27434 −0.637171 0.770723i \(-0.719895\pi\)
−0.637171 + 0.770723i \(0.719895\pi\)
\(198\) 5.35093e7 0.489893
\(199\) −4.28301e7 −0.385268 −0.192634 0.981271i \(-0.561703\pi\)
−0.192634 + 0.981271i \(0.561703\pi\)
\(200\) −5.36689e8 −4.74371
\(201\) 6.91841e7 0.600925
\(202\) 8.57040e7 0.731596
\(203\) −4.58727e7 −0.384874
\(204\) 1.60237e7 0.132147
\(205\) 3.38892e8 2.74740
\(206\) 3.43509e8 2.73781
\(207\) 6.83444e6 0.0535558
\(208\) −1.92806e8 −1.48559
\(209\) −2.23716e7 −0.169506
\(210\) −7.56090e7 −0.563386
\(211\) −4.75756e7 −0.348655 −0.174327 0.984688i \(-0.555775\pi\)
−0.174327 + 0.984688i \(0.555775\pi\)
\(212\) −4.67864e8 −3.37244
\(213\) −8.55486e7 −0.606575
\(214\) 3.00602e8 2.09673
\(215\) 3.31982e8 2.27814
\(216\) 7.61691e7 0.514269
\(217\) −7.80350e7 −0.518419
\(218\) 1.39805e7 0.0913958
\(219\) −4.39992e7 −0.283067
\(220\) 5.08781e8 3.22145
\(221\) 8.88517e6 0.0553723
\(222\) −2.00252e7 −0.122840
\(223\) 1.14043e8 0.688652 0.344326 0.938850i \(-0.388107\pi\)
0.344326 + 0.938850i \(0.388107\pi\)
\(224\) −1.05966e8 −0.629940
\(225\) 1.01103e8 0.591731
\(226\) −4.99431e8 −2.87803
\(227\) 5.49344e7 0.311712 0.155856 0.987780i \(-0.450186\pi\)
0.155856 + 0.987780i \(0.450186\pi\)
\(228\) −5.39506e7 −0.301456
\(229\) −5.39847e7 −0.297062 −0.148531 0.988908i \(-0.547454\pi\)
−0.148531 + 0.988908i \(0.547454\pi\)
\(230\) 9.16096e7 0.496471
\(231\) 2.70636e7 0.144459
\(232\) 6.19437e8 3.25679
\(233\) −2.85964e8 −1.48104 −0.740518 0.672036i \(-0.765420\pi\)
−0.740518 + 0.672036i \(0.765420\pi\)
\(234\) 7.15536e7 0.365070
\(235\) 2.21925e8 1.11550
\(236\) −6.41606e7 −0.317743
\(237\) 5.70177e7 0.278221
\(238\) 1.14250e7 0.0549332
\(239\) −8.27309e7 −0.391990 −0.195995 0.980605i \(-0.562794\pi\)
−0.195995 + 0.980605i \(0.562794\pi\)
\(240\) 5.18257e8 2.41995
\(241\) −2.15524e8 −0.991825 −0.495912 0.868373i \(-0.665166\pi\)
−0.495912 + 0.868373i \(0.665166\pi\)
\(242\) 1.52221e8 0.690433
\(243\) −1.43489e7 −0.0641500
\(244\) −3.27262e8 −1.44222
\(245\) 3.45226e8 1.49976
\(246\) −4.12389e8 −1.76618
\(247\) −2.99157e7 −0.126317
\(248\) 1.05374e9 4.38684
\(249\) −1.46821e8 −0.602685
\(250\) 5.91786e8 2.39538
\(251\) −3.78187e8 −1.50955 −0.754776 0.655982i \(-0.772255\pi\)
−0.754776 + 0.655982i \(0.772255\pi\)
\(252\) 6.52656e7 0.256911
\(253\) −3.27909e7 −0.127301
\(254\) −1.63373e8 −0.625551
\(255\) −2.38831e7 −0.0901988
\(256\) −2.17495e8 −0.810233
\(257\) 2.36018e8 0.867320 0.433660 0.901076i \(-0.357222\pi\)
0.433660 + 0.901076i \(0.357222\pi\)
\(258\) −4.03982e8 −1.46451
\(259\) −1.01282e7 −0.0362229
\(260\) 6.80352e8 2.40064
\(261\) −1.16691e8 −0.406252
\(262\) 5.63843e8 1.93688
\(263\) −3.99714e8 −1.35489 −0.677445 0.735573i \(-0.736913\pi\)
−0.677445 + 0.735573i \(0.736913\pi\)
\(264\) −3.65450e8 −1.22240
\(265\) 6.97346e8 2.30191
\(266\) −3.84670e7 −0.125315
\(267\) 6.14262e7 0.197499
\(268\) −8.00489e8 −2.54029
\(269\) 3.97589e8 1.24538 0.622689 0.782470i \(-0.286040\pi\)
0.622689 + 0.782470i \(0.286040\pi\)
\(270\) −1.92334e8 −0.594681
\(271\) 5.14541e8 1.57046 0.785232 0.619202i \(-0.212544\pi\)
0.785232 + 0.619202i \(0.212544\pi\)
\(272\) −7.83116e7 −0.235958
\(273\) 3.61899e7 0.107651
\(274\) 1.05441e9 3.09658
\(275\) −4.85079e8 −1.40653
\(276\) −7.90773e7 −0.226397
\(277\) −2.94631e8 −0.832911 −0.416455 0.909156i \(-0.636728\pi\)
−0.416455 + 0.909156i \(0.636728\pi\)
\(278\) 1.14777e9 3.20405
\(279\) −1.98506e8 −0.547215
\(280\) 5.16384e8 1.40579
\(281\) 6.15936e8 1.65601 0.828006 0.560719i \(-0.189475\pi\)
0.828006 + 0.560719i \(0.189475\pi\)
\(282\) −2.70055e8 −0.717102
\(283\) −4.48859e6 −0.0117722 −0.00588609 0.999983i \(-0.501874\pi\)
−0.00588609 + 0.999983i \(0.501874\pi\)
\(284\) 9.89832e8 2.56417
\(285\) 8.04128e7 0.205764
\(286\) −3.43306e8 −0.867761
\(287\) −2.08576e8 −0.520807
\(288\) −2.69557e8 −0.664932
\(289\) −4.06730e8 −0.991205
\(290\) −1.56414e9 −3.76602
\(291\) −2.70071e8 −0.642470
\(292\) 5.09088e8 1.19661
\(293\) 3.57368e8 0.830002 0.415001 0.909821i \(-0.363781\pi\)
0.415001 + 0.909821i \(0.363781\pi\)
\(294\) −4.20097e8 −0.964126
\(295\) 9.56307e7 0.216880
\(296\) 1.36765e8 0.306517
\(297\) 6.88444e7 0.152483
\(298\) −7.79390e8 −1.70607
\(299\) −4.38485e7 −0.0948650
\(300\) −1.16980e9 −2.50142
\(301\) −2.04323e8 −0.431852
\(302\) −2.86457e8 −0.598461
\(303\) 1.10266e8 0.227715
\(304\) 2.63670e8 0.538273
\(305\) 4.87780e8 0.984407
\(306\) 2.90628e7 0.0579846
\(307\) −4.17291e8 −0.823105 −0.411552 0.911386i \(-0.635013\pi\)
−0.411552 + 0.911386i \(0.635013\pi\)
\(308\) −3.13137e8 −0.610670
\(309\) 4.41955e8 0.852164
\(310\) −2.66079e9 −5.07277
\(311\) 1.57129e8 0.296207 0.148103 0.988972i \(-0.452683\pi\)
0.148103 + 0.988972i \(0.452683\pi\)
\(312\) −4.88687e8 −0.910940
\(313\) 8.52228e8 1.57091 0.785454 0.618920i \(-0.212430\pi\)
0.785454 + 0.618920i \(0.212430\pi\)
\(314\) −7.92454e8 −1.44451
\(315\) −9.72777e7 −0.175358
\(316\) −6.59718e8 −1.17612
\(317\) −7.40371e8 −1.30539 −0.652697 0.757619i \(-0.726363\pi\)
−0.652697 + 0.757619i \(0.726363\pi\)
\(318\) −8.48584e8 −1.47979
\(319\) 5.59870e8 0.965651
\(320\) −1.15625e9 −1.97254
\(321\) 3.86751e8 0.652625
\(322\) −5.63824e7 −0.0941127
\(323\) −1.21508e7 −0.0200631
\(324\) 1.66023e8 0.271181
\(325\) −6.48657e8 −1.04815
\(326\) −1.03361e9 −1.65232
\(327\) 1.79872e7 0.0284476
\(328\) 2.81648e9 4.40705
\(329\) −1.36587e8 −0.211458
\(330\) 9.22798e8 1.41354
\(331\) −2.16470e8 −0.328095 −0.164047 0.986452i \(-0.552455\pi\)
−0.164047 + 0.986452i \(0.552455\pi\)
\(332\) 1.69878e9 2.54773
\(333\) −2.57642e7 −0.0382350
\(334\) −2.18642e9 −3.21085
\(335\) 1.19312e9 1.73391
\(336\) −3.18969e8 −0.458734
\(337\) −1.73965e8 −0.247604 −0.123802 0.992307i \(-0.539509\pi\)
−0.123802 + 0.992307i \(0.539509\pi\)
\(338\) 8.57749e8 1.20824
\(339\) −6.42562e8 −0.895810
\(340\) 2.76338e8 0.381297
\(341\) 9.52407e8 1.30072
\(342\) −9.78525e7 −0.132276
\(343\) −4.48484e8 −0.600092
\(344\) 2.75906e9 3.65432
\(345\) 1.17864e8 0.154530
\(346\) −2.01606e9 −2.61659
\(347\) −7.38996e7 −0.0949486 −0.0474743 0.998872i \(-0.515117\pi\)
−0.0474743 + 0.998872i \(0.515117\pi\)
\(348\) 1.35016e9 1.71735
\(349\) 5.44309e8 0.685419 0.342710 0.939441i \(-0.388655\pi\)
0.342710 + 0.939441i \(0.388655\pi\)
\(350\) −8.34072e8 −1.03984
\(351\) 9.20600e7 0.113631
\(352\) 1.29330e9 1.58052
\(353\) 1.58939e9 1.92317 0.961587 0.274499i \(-0.0885119\pi\)
0.961587 + 0.274499i \(0.0885119\pi\)
\(354\) −1.16371e8 −0.139422
\(355\) −1.47533e9 −1.75021
\(356\) −7.10726e8 −0.834886
\(357\) 1.46992e7 0.0170984
\(358\) 1.79358e9 2.06599
\(359\) 1.69409e9 1.93244 0.966221 0.257713i \(-0.0829689\pi\)
0.966221 + 0.257713i \(0.0829689\pi\)
\(360\) 1.31358e9 1.48388
\(361\) −8.52961e8 −0.954232
\(362\) −8.68914e8 −0.962713
\(363\) 1.95846e8 0.214903
\(364\) −4.18732e8 −0.455073
\(365\) −7.58791e8 −0.816765
\(366\) −5.93568e8 −0.632830
\(367\) −1.50784e9 −1.59230 −0.796151 0.605098i \(-0.793134\pi\)
−0.796151 + 0.605098i \(0.793134\pi\)
\(368\) 3.86470e8 0.404248
\(369\) −5.30575e8 −0.549737
\(370\) −3.45346e8 −0.354445
\(371\) −4.29192e8 −0.436358
\(372\) 2.29679e9 2.31324
\(373\) −9.09574e8 −0.907522 −0.453761 0.891123i \(-0.649918\pi\)
−0.453761 + 0.891123i \(0.649918\pi\)
\(374\) −1.39440e8 −0.137828
\(375\) 7.61385e8 0.745581
\(376\) 1.84439e9 1.78935
\(377\) 7.48669e8 0.719606
\(378\) 1.18375e8 0.112730
\(379\) 1.38013e9 1.30221 0.651106 0.758987i \(-0.274305\pi\)
0.651106 + 0.758987i \(0.274305\pi\)
\(380\) −9.30409e8 −0.869824
\(381\) −2.10194e8 −0.194708
\(382\) 1.67958e9 1.54163
\(383\) −1.20160e9 −1.09286 −0.546430 0.837505i \(-0.684013\pi\)
−0.546430 + 0.837505i \(0.684013\pi\)
\(384\) 1.29113e8 0.116362
\(385\) 4.66727e8 0.416822
\(386\) −2.85799e9 −2.52933
\(387\) −5.19758e8 −0.455840
\(388\) 3.12483e9 2.71591
\(389\) 1.11547e9 0.960802 0.480401 0.877049i \(-0.340491\pi\)
0.480401 + 0.877049i \(0.340491\pi\)
\(390\) 1.23398e9 1.05337
\(391\) −1.78099e7 −0.0150676
\(392\) 2.86912e9 2.40574
\(393\) 7.25434e8 0.602870
\(394\) 2.86973e9 2.36377
\(395\) 9.83303e8 0.802782
\(396\) −7.96558e8 −0.644591
\(397\) −1.21715e9 −0.976286 −0.488143 0.872764i \(-0.662325\pi\)
−0.488143 + 0.872764i \(0.662325\pi\)
\(398\) 8.98820e8 0.714631
\(399\) −4.94912e7 −0.0390052
\(400\) 5.71709e9 4.46648
\(401\) −9.38458e8 −0.726791 −0.363396 0.931635i \(-0.618383\pi\)
−0.363396 + 0.931635i \(0.618383\pi\)
\(402\) −1.45188e9 −1.11465
\(403\) 1.27358e9 0.969298
\(404\) −1.27582e9 −0.962620
\(405\) −2.47455e8 −0.185099
\(406\) 9.62672e8 0.713900
\(407\) 1.23614e8 0.0908836
\(408\) −1.98489e8 −0.144686
\(409\) −2.00800e9 −1.45122 −0.725608 0.688108i \(-0.758441\pi\)
−0.725608 + 0.688108i \(0.758441\pi\)
\(410\) −7.11189e9 −5.09614
\(411\) 1.35659e9 0.963836
\(412\) −5.11360e9 −3.60235
\(413\) −5.88573e7 −0.0411126
\(414\) −1.43426e8 −0.0993404
\(415\) −2.53201e9 −1.73899
\(416\) 1.72943e9 1.17781
\(417\) 1.47671e9 0.997286
\(418\) 4.69485e8 0.314416
\(419\) −2.40847e9 −1.59953 −0.799765 0.600313i \(-0.795043\pi\)
−0.799765 + 0.600313i \(0.795043\pi\)
\(420\) 1.12554e9 0.741292
\(421\) −5.54976e8 −0.362482 −0.181241 0.983439i \(-0.558011\pi\)
−0.181241 + 0.983439i \(0.558011\pi\)
\(422\) 9.98409e8 0.646718
\(423\) −3.47450e8 −0.223204
\(424\) 5.79555e9 3.69244
\(425\) −2.63464e8 −0.166479
\(426\) 1.79530e9 1.12513
\(427\) −3.00211e8 −0.186608
\(428\) −4.47486e9 −2.75884
\(429\) −4.41693e8 −0.270097
\(430\) −6.96690e9 −4.22571
\(431\) −2.19055e9 −1.31790 −0.658951 0.752186i \(-0.728999\pi\)
−0.658951 + 0.752186i \(0.728999\pi\)
\(432\) −8.11393e8 −0.484215
\(433\) 2.56663e8 0.151934 0.0759672 0.997110i \(-0.475796\pi\)
0.0759672 + 0.997110i \(0.475796\pi\)
\(434\) 1.63762e9 0.961611
\(435\) −2.01241e9 −1.17220
\(436\) −2.08119e8 −0.120257
\(437\) 5.99647e7 0.0343725
\(438\) 9.23355e8 0.525060
\(439\) 2.12343e9 1.19787 0.598937 0.800796i \(-0.295590\pi\)
0.598937 + 0.800796i \(0.295590\pi\)
\(440\) −6.30240e9 −3.52713
\(441\) −5.40492e8 −0.300092
\(442\) −1.86462e8 −0.102710
\(443\) 3.50552e9 1.91575 0.957875 0.287185i \(-0.0927193\pi\)
0.957875 + 0.287185i \(0.0927193\pi\)
\(444\) 2.98102e8 0.161631
\(445\) 1.05933e9 0.569864
\(446\) −2.39327e9 −1.27738
\(447\) −1.00275e9 −0.531029
\(448\) 7.11630e8 0.373922
\(449\) 1.44417e9 0.752933 0.376467 0.926430i \(-0.377139\pi\)
0.376467 + 0.926430i \(0.377139\pi\)
\(450\) −2.12171e9 −1.09760
\(451\) 2.54564e9 1.30671
\(452\) 7.43470e9 3.78686
\(453\) −3.68553e8 −0.186276
\(454\) −1.15284e9 −0.578193
\(455\) 6.24116e8 0.310617
\(456\) 6.68300e8 0.330061
\(457\) −3.55204e7 −0.0174089 −0.00870445 0.999962i \(-0.502771\pi\)
−0.00870445 + 0.999962i \(0.502771\pi\)
\(458\) 1.13291e9 0.551018
\(459\) 3.73919e7 0.0180482
\(460\) −1.36373e9 −0.653246
\(461\) −1.47773e9 −0.702491 −0.351245 0.936283i \(-0.614242\pi\)
−0.351245 + 0.936283i \(0.614242\pi\)
\(462\) −5.67949e8 −0.267955
\(463\) −1.25958e9 −0.589782 −0.294891 0.955531i \(-0.595283\pi\)
−0.294891 + 0.955531i \(0.595283\pi\)
\(464\) −6.59858e9 −3.06646
\(465\) −3.42335e9 −1.57894
\(466\) 6.00117e9 2.74717
\(467\) 1.25254e9 0.569090 0.284545 0.958663i \(-0.408157\pi\)
0.284545 + 0.958663i \(0.408157\pi\)
\(468\) −1.06517e9 −0.480351
\(469\) −7.34323e8 −0.328687
\(470\) −4.65726e9 −2.06913
\(471\) −1.01956e9 −0.449614
\(472\) 7.94773e8 0.347894
\(473\) 2.49374e9 1.08352
\(474\) −1.19656e9 −0.516071
\(475\) 8.87065e8 0.379776
\(476\) −1.70076e8 −0.0722800
\(477\) −1.09178e9 −0.460596
\(478\) 1.73617e9 0.727100
\(479\) 3.28427e9 1.36541 0.682707 0.730693i \(-0.260803\pi\)
0.682707 + 0.730693i \(0.260803\pi\)
\(480\) −4.64867e9 −1.91860
\(481\) 1.65298e8 0.0677268
\(482\) 4.52292e9 1.83973
\(483\) −7.25410e7 −0.0292933
\(484\) −2.26602e9 −0.908458
\(485\) −4.65753e9 −1.85379
\(486\) 3.01122e8 0.118992
\(487\) 1.74837e8 0.0685935 0.0342968 0.999412i \(-0.489081\pi\)
0.0342968 + 0.999412i \(0.489081\pi\)
\(488\) 4.05387e9 1.57907
\(489\) −1.32983e9 −0.514298
\(490\) −7.24482e9 −2.78190
\(491\) −1.05796e9 −0.403353 −0.201676 0.979452i \(-0.564639\pi\)
−0.201676 + 0.979452i \(0.564639\pi\)
\(492\) 6.13897e9 2.32390
\(493\) 3.04086e8 0.114296
\(494\) 6.27804e8 0.234304
\(495\) 1.18726e9 0.439975
\(496\) −1.12250e10 −4.13047
\(497\) 9.08015e8 0.331777
\(498\) 3.08115e9 1.11792
\(499\) −7.20840e8 −0.259709 −0.129854 0.991533i \(-0.541451\pi\)
−0.129854 + 0.991533i \(0.541451\pi\)
\(500\) −8.80954e9 −3.15180
\(501\) −2.81302e9 −0.999402
\(502\) 7.93653e9 2.80006
\(503\) 5.82049e8 0.203925 0.101963 0.994788i \(-0.467488\pi\)
0.101963 + 0.994788i \(0.467488\pi\)
\(504\) −8.08461e8 −0.281289
\(505\) 1.90160e9 0.657051
\(506\) 6.88140e8 0.236129
\(507\) 1.10357e9 0.376073
\(508\) 2.43203e9 0.823088
\(509\) −5.21497e9 −1.75283 −0.876414 0.481558i \(-0.840071\pi\)
−0.876414 + 0.481558i \(0.840071\pi\)
\(510\) 5.01205e8 0.167309
\(511\) 4.67009e8 0.154829
\(512\) 5.17639e9 1.70444
\(513\) −1.25896e8 −0.0411719
\(514\) −4.95301e9 −1.60879
\(515\) 7.62177e9 2.45884
\(516\) 6.01381e9 1.92697
\(517\) 1.66703e9 0.530549
\(518\) 2.12548e8 0.0671897
\(519\) −2.59383e9 −0.814435
\(520\) −8.42769e9 −2.62843
\(521\) 4.72585e8 0.146402 0.0732011 0.997317i \(-0.476679\pi\)
0.0732011 + 0.997317i \(0.476679\pi\)
\(522\) 2.44885e9 0.753555
\(523\) 8.79553e8 0.268847 0.134424 0.990924i \(-0.457082\pi\)
0.134424 + 0.990924i \(0.457082\pi\)
\(524\) −8.39357e9 −2.54851
\(525\) −1.07311e9 −0.323658
\(526\) 8.38829e9 2.51318
\(527\) 5.17287e8 0.153955
\(528\) 3.89297e9 1.15097
\(529\) −3.31693e9 −0.974186
\(530\) −1.46343e10 −4.26980
\(531\) −1.49721e8 −0.0433963
\(532\) 5.72633e8 0.164887
\(533\) 3.40407e9 0.973764
\(534\) −1.28907e9 −0.366339
\(535\) 6.66974e9 1.88309
\(536\) 9.91585e9 2.78134
\(537\) 2.30759e9 0.643057
\(538\) −8.34369e9 −2.31004
\(539\) 2.59322e9 0.713310
\(540\) 2.86316e9 0.782469
\(541\) −1.83033e8 −0.0496981 −0.0248491 0.999691i \(-0.507911\pi\)
−0.0248491 + 0.999691i \(0.507911\pi\)
\(542\) −1.07980e10 −2.91304
\(543\) −1.11793e9 −0.299652
\(544\) 7.02440e8 0.187074
\(545\) 3.10200e8 0.0820830
\(546\) −7.59472e8 −0.199681
\(547\) −2.64278e9 −0.690408 −0.345204 0.938528i \(-0.612190\pi\)
−0.345204 + 0.938528i \(0.612190\pi\)
\(548\) −1.56963e10 −4.07442
\(549\) −7.63678e8 −0.196973
\(550\) 1.01797e10 2.60896
\(551\) −1.02384e9 −0.260735
\(552\) 9.79550e8 0.247879
\(553\) −6.05187e8 −0.152178
\(554\) 6.18304e9 1.54496
\(555\) −4.44318e8 −0.110324
\(556\) −1.70862e10 −4.21583
\(557\) 2.28927e9 0.561313 0.280656 0.959808i \(-0.409448\pi\)
0.280656 + 0.959808i \(0.409448\pi\)
\(558\) 4.16579e9 1.01503
\(559\) 3.33467e9 0.807443
\(560\) −5.50080e9 −1.32363
\(561\) −1.79402e8 −0.0429000
\(562\) −1.29259e10 −3.07173
\(563\) 1.07781e9 0.254543 0.127271 0.991868i \(-0.459378\pi\)
0.127271 + 0.991868i \(0.459378\pi\)
\(564\) 4.02014e9 0.943548
\(565\) −1.10814e10 −2.58478
\(566\) 9.41963e7 0.0218362
\(567\) 1.52300e8 0.0350880
\(568\) −1.22613e10 −2.80748
\(569\) −1.08728e9 −0.247429 −0.123714 0.992318i \(-0.539481\pi\)
−0.123714 + 0.992318i \(0.539481\pi\)
\(570\) −1.68752e9 −0.381670
\(571\) 2.15499e8 0.0484416 0.0242208 0.999707i \(-0.492290\pi\)
0.0242208 + 0.999707i \(0.492290\pi\)
\(572\) 5.11057e9 1.14178
\(573\) 2.16093e9 0.479844
\(574\) 4.37711e9 0.966042
\(575\) 1.30020e9 0.285216
\(576\) 1.81025e9 0.394693
\(577\) 7.86457e9 1.70435 0.852176 0.523255i \(-0.175282\pi\)
0.852176 + 0.523255i \(0.175282\pi\)
\(578\) 8.53552e9 1.83858
\(579\) −3.67705e9 −0.787273
\(580\) 2.32843e10 4.95526
\(581\) 1.55836e9 0.329649
\(582\) 5.66764e9 1.19171
\(583\) 5.23823e9 1.09482
\(584\) −6.30620e9 −1.31016
\(585\) 1.58763e9 0.327871
\(586\) −7.49963e9 −1.53957
\(587\) 7.86924e9 1.60583 0.802914 0.596094i \(-0.203282\pi\)
0.802914 + 0.596094i \(0.203282\pi\)
\(588\) 6.25371e9 1.26858
\(589\) −1.74167e9 −0.351206
\(590\) −2.00688e9 −0.402290
\(591\) 3.69217e9 0.735741
\(592\) −1.45690e9 −0.288604
\(593\) −9.64856e9 −1.90008 −0.950038 0.312134i \(-0.898956\pi\)
−0.950038 + 0.312134i \(0.898956\pi\)
\(594\) −1.44475e9 −0.282840
\(595\) 2.53496e8 0.0493358
\(596\) 1.16023e10 2.24482
\(597\) 1.15641e9 0.222435
\(598\) 9.20194e8 0.175965
\(599\) 5.43654e9 1.03354 0.516772 0.856123i \(-0.327134\pi\)
0.516772 + 0.856123i \(0.327134\pi\)
\(600\) 1.44906e10 2.73878
\(601\) −2.28199e9 −0.428798 −0.214399 0.976746i \(-0.568779\pi\)
−0.214399 + 0.976746i \(0.568779\pi\)
\(602\) 4.28787e9 0.801040
\(603\) −1.86797e9 −0.346944
\(604\) 4.26431e9 0.787443
\(605\) 3.37748e9 0.620082
\(606\) −2.31401e9 −0.422387
\(607\) −2.22820e9 −0.404383 −0.202191 0.979346i \(-0.564806\pi\)
−0.202191 + 0.979346i \(0.564806\pi\)
\(608\) −2.36507e9 −0.426757
\(609\) 1.23856e9 0.222207
\(610\) −1.02364e10 −1.82597
\(611\) 2.22918e9 0.395367
\(612\) −4.32639e8 −0.0762950
\(613\) 9.10899e8 0.159720 0.0798599 0.996806i \(-0.474553\pi\)
0.0798599 + 0.996806i \(0.474553\pi\)
\(614\) 8.75716e9 1.52677
\(615\) −9.15008e9 −1.58621
\(616\) 3.87890e9 0.668615
\(617\) 6.17621e9 1.05858 0.529290 0.848441i \(-0.322458\pi\)
0.529290 + 0.848441i \(0.322458\pi\)
\(618\) −9.27475e9 −1.58067
\(619\) −9.92262e9 −1.68155 −0.840774 0.541387i \(-0.817900\pi\)
−0.840774 + 0.541387i \(0.817900\pi\)
\(620\) 3.96095e10 6.67465
\(621\) −1.84530e8 −0.0309205
\(622\) −3.29747e9 −0.549432
\(623\) −6.51980e8 −0.108025
\(624\) 5.20575e9 0.857704
\(625\) 2.29562e9 0.376114
\(626\) −1.78846e10 −2.91387
\(627\) 6.04034e8 0.0978644
\(628\) 1.17967e10 1.90066
\(629\) 6.71390e7 0.0107572
\(630\) 2.04144e9 0.325271
\(631\) 3.99548e9 0.633091 0.316546 0.948577i \(-0.397477\pi\)
0.316546 + 0.948577i \(0.397477\pi\)
\(632\) 8.17209e9 1.28773
\(633\) 1.28454e9 0.201296
\(634\) 1.55372e10 2.42137
\(635\) −3.62492e9 −0.561811
\(636\) 1.26323e10 1.94708
\(637\) 3.46770e9 0.531561
\(638\) −1.17493e10 −1.79118
\(639\) 2.30981e9 0.350206
\(640\) 2.22663e9 0.335752
\(641\) −4.03744e9 −0.605485 −0.302742 0.953072i \(-0.597902\pi\)
−0.302742 + 0.953072i \(0.597902\pi\)
\(642\) −8.11624e9 −1.21055
\(643\) −8.64628e9 −1.28260 −0.641299 0.767291i \(-0.721604\pi\)
−0.641299 + 0.767291i \(0.721604\pi\)
\(644\) 8.39329e8 0.123832
\(645\) −8.96353e9 −1.31529
\(646\) 2.54994e8 0.0372149
\(647\) 7.61025e9 1.10467 0.552337 0.833621i \(-0.313736\pi\)
0.552337 + 0.833621i \(0.313736\pi\)
\(648\) −2.05656e9 −0.296913
\(649\) 7.18346e8 0.103152
\(650\) 1.36125e10 1.94421
\(651\) 2.10695e9 0.299309
\(652\) 1.53867e10 2.17409
\(653\) −1.04681e10 −1.47120 −0.735599 0.677417i \(-0.763099\pi\)
−0.735599 + 0.677417i \(0.763099\pi\)
\(654\) −3.77474e8 −0.0527674
\(655\) 1.25105e10 1.73953
\(656\) −3.00026e10 −4.14950
\(657\) 1.18798e9 0.163429
\(658\) 2.86638e9 0.392232
\(659\) 3.31882e9 0.451737 0.225868 0.974158i \(-0.427478\pi\)
0.225868 + 0.974158i \(0.427478\pi\)
\(660\) −1.37371e10 −1.85991
\(661\) 7.95737e9 1.07168 0.535839 0.844320i \(-0.319995\pi\)
0.535839 + 0.844320i \(0.319995\pi\)
\(662\) 4.54278e9 0.608581
\(663\) −2.39900e8 −0.0319692
\(664\) −2.10432e10 −2.78948
\(665\) −8.53504e8 −0.112546
\(666\) 5.40680e8 0.0709219
\(667\) −1.50067e9 −0.195815
\(668\) 3.25478e10 4.22477
\(669\) −3.07915e9 −0.397593
\(670\) −2.50385e10 −3.21623
\(671\) 3.66404e9 0.468200
\(672\) 2.86109e9 0.363696
\(673\) −7.14650e9 −0.903734 −0.451867 0.892085i \(-0.649242\pi\)
−0.451867 + 0.892085i \(0.649242\pi\)
\(674\) 3.65079e9 0.459279
\(675\) −2.72977e9 −0.341636
\(676\) −1.27687e10 −1.58977
\(677\) −7.35419e9 −0.910908 −0.455454 0.890259i \(-0.650523\pi\)
−0.455454 + 0.890259i \(0.650523\pi\)
\(678\) 1.34846e10 1.66163
\(679\) 2.86654e9 0.351410
\(680\) −3.42306e9 −0.417478
\(681\) −1.48323e9 −0.179967
\(682\) −1.99870e10 −2.41269
\(683\) 4.04823e9 0.486175 0.243087 0.970004i \(-0.421840\pi\)
0.243087 + 0.970004i \(0.421840\pi\)
\(684\) 1.45667e9 0.174046
\(685\) 2.33952e10 2.78106
\(686\) 9.41177e9 1.11311
\(687\) 1.45759e9 0.171509
\(688\) −2.93910e10 −3.44076
\(689\) 7.00465e9 0.815867
\(690\) −2.47346e9 −0.286638
\(691\) −1.70309e9 −0.196365 −0.0981823 0.995168i \(-0.531303\pi\)
−0.0981823 + 0.995168i \(0.531303\pi\)
\(692\) 3.00117e10 3.44286
\(693\) −7.30717e8 −0.0834032
\(694\) 1.55084e9 0.176120
\(695\) 2.54668e10 2.87758
\(696\) −1.67248e10 −1.88031
\(697\) 1.38263e9 0.154665
\(698\) −1.14227e10 −1.27138
\(699\) 7.72103e9 0.855077
\(700\) 1.24163e10 1.36820
\(701\) 7.38709e9 0.809954 0.404977 0.914327i \(-0.367279\pi\)
0.404977 + 0.914327i \(0.367279\pi\)
\(702\) −1.93195e9 −0.210773
\(703\) −2.26052e8 −0.0245395
\(704\) −8.68536e9 −0.938174
\(705\) −5.99198e9 −0.644033
\(706\) −3.33545e10 −3.56729
\(707\) −1.17037e9 −0.124553
\(708\) 1.73234e9 0.183449
\(709\) −3.00194e9 −0.316330 −0.158165 0.987413i \(-0.550558\pi\)
−0.158165 + 0.987413i \(0.550558\pi\)
\(710\) 3.09610e10 3.24646
\(711\) −1.53948e9 −0.160631
\(712\) 8.80394e9 0.914108
\(713\) −2.55282e9 −0.263759
\(714\) −3.08474e8 −0.0317157
\(715\) −7.61726e9 −0.779341
\(716\) −2.66998e10 −2.71839
\(717\) 2.23373e9 0.226316
\(718\) −3.55518e10 −3.58448
\(719\) 3.86807e9 0.388100 0.194050 0.980992i \(-0.437838\pi\)
0.194050 + 0.980992i \(0.437838\pi\)
\(720\) −1.39929e10 −1.39716
\(721\) −4.69092e9 −0.466106
\(722\) 1.79000e10 1.77000
\(723\) 5.81913e9 0.572630
\(724\) 1.29350e10 1.26672
\(725\) −2.21996e10 −2.16353
\(726\) −4.10997e9 −0.398622
\(727\) 1.88812e10 1.82247 0.911235 0.411887i \(-0.135130\pi\)
0.911235 + 0.411887i \(0.135130\pi\)
\(728\) 5.18694e9 0.498255
\(729\) 3.87420e8 0.0370370
\(730\) 1.59238e10 1.51501
\(731\) 1.35444e9 0.128247
\(732\) 8.83606e9 0.832665
\(733\) −2.89464e9 −0.271476 −0.135738 0.990745i \(-0.543340\pi\)
−0.135738 + 0.990745i \(0.543340\pi\)
\(734\) 3.16432e10 2.95355
\(735\) −9.32110e9 −0.865887
\(736\) −3.46656e9 −0.320499
\(737\) 8.96231e9 0.824677
\(738\) 1.11345e10 1.01970
\(739\) −4.83541e9 −0.440735 −0.220368 0.975417i \(-0.570726\pi\)
−0.220368 + 0.975417i \(0.570726\pi\)
\(740\) 5.14094e9 0.466371
\(741\) 8.07725e8 0.0729289
\(742\) 9.00690e9 0.809397
\(743\) −1.65827e10 −1.48318 −0.741591 0.670852i \(-0.765929\pi\)
−0.741591 + 0.670852i \(0.765929\pi\)
\(744\) −2.84509e10 −2.53274
\(745\) −1.72931e10 −1.53223
\(746\) 1.90881e10 1.68336
\(747\) 3.96417e9 0.347961
\(748\) 2.07575e9 0.181351
\(749\) −4.10498e9 −0.356965
\(750\) −1.59782e10 −1.38297
\(751\) 1.94126e10 1.67242 0.836208 0.548413i \(-0.184768\pi\)
0.836208 + 0.548413i \(0.184768\pi\)
\(752\) −1.96474e10 −1.68478
\(753\) 1.02110e10 0.871541
\(754\) −1.57114e10 −1.33479
\(755\) −6.35591e9 −0.537481
\(756\) −1.76217e9 −0.148327
\(757\) 1.23613e10 1.03569 0.517843 0.855475i \(-0.326735\pi\)
0.517843 + 0.855475i \(0.326735\pi\)
\(758\) −2.89630e10 −2.41547
\(759\) 8.85353e8 0.0734971
\(760\) 1.15252e10 0.952361
\(761\) −1.27885e10 −1.05189 −0.525947 0.850517i \(-0.676289\pi\)
−0.525947 + 0.850517i \(0.676289\pi\)
\(762\) 4.41108e9 0.361162
\(763\) −1.90917e8 −0.0155599
\(764\) −2.50029e10 −2.02845
\(765\) 6.44845e8 0.0520763
\(766\) 2.52165e10 2.02714
\(767\) 9.60585e8 0.0768691
\(768\) 5.87238e9 0.467788
\(769\) −6.74352e9 −0.534742 −0.267371 0.963594i \(-0.586155\pi\)
−0.267371 + 0.963594i \(0.586155\pi\)
\(770\) −9.79461e9 −0.773160
\(771\) −6.37249e9 −0.500748
\(772\) 4.25450e10 3.32804
\(773\) 3.73822e9 0.291096 0.145548 0.989351i \(-0.453505\pi\)
0.145548 + 0.989351i \(0.453505\pi\)
\(774\) 1.09075e10 0.845536
\(775\) −3.77642e10 −2.91424
\(776\) −3.87081e10 −2.97362
\(777\) 2.73462e8 0.0209133
\(778\) −2.34089e10 −1.78219
\(779\) −4.65521e9 −0.352824
\(780\) −1.83695e10 −1.38601
\(781\) −1.10822e10 −0.832430
\(782\) 3.73754e8 0.0279487
\(783\) 3.15066e9 0.234550
\(784\) −3.05634e10 −2.26514
\(785\) −1.75829e10 −1.29732
\(786\) −1.52238e10 −1.11826
\(787\) −1.47464e10 −1.07839 −0.539193 0.842182i \(-0.681271\pi\)
−0.539193 + 0.842182i \(0.681271\pi\)
\(788\) −4.27199e10 −3.11020
\(789\) 1.07923e10 0.782246
\(790\) −2.06353e10 −1.48907
\(791\) 6.82017e9 0.489979
\(792\) 9.86716e9 0.705755
\(793\) 4.89962e9 0.348904
\(794\) 2.55428e10 1.81091
\(795\) −1.88284e10 −1.32901
\(796\) −1.33802e10 −0.940298
\(797\) −1.86115e10 −1.30220 −0.651099 0.758993i \(-0.725692\pi\)
−0.651099 + 0.758993i \(0.725692\pi\)
\(798\) 1.03861e9 0.0723506
\(799\) 9.05422e8 0.0627968
\(800\) −5.12812e10 −3.54114
\(801\) −1.65851e9 −0.114026
\(802\) 1.96942e10 1.34812
\(803\) −5.69978e9 −0.388466
\(804\) 2.16132e10 1.46664
\(805\) −1.25101e9 −0.0845231
\(806\) −2.67269e10 −1.79795
\(807\) −1.07349e10 −0.719019
\(808\) 1.58039e10 1.05396
\(809\) 4.53515e9 0.301143 0.150571 0.988599i \(-0.451889\pi\)
0.150571 + 0.988599i \(0.451889\pi\)
\(810\) 5.19303e9 0.343339
\(811\) 7.48809e9 0.492945 0.246472 0.969150i \(-0.420729\pi\)
0.246472 + 0.969150i \(0.420729\pi\)
\(812\) −1.43307e10 −0.939335
\(813\) −1.38926e10 −0.906707
\(814\) −2.59412e9 −0.168579
\(815\) −2.29337e10 −1.48396
\(816\) 2.11441e9 0.136230
\(817\) −4.56030e9 −0.292561
\(818\) 4.21393e10 2.69185
\(819\) −9.77128e8 −0.0621524
\(820\) 1.05870e11 6.70540
\(821\) −1.66654e10 −1.05103 −0.525514 0.850785i \(-0.676127\pi\)
−0.525514 + 0.850785i \(0.676127\pi\)
\(822\) −2.84691e10 −1.78781
\(823\) 1.72880e10 1.08105 0.540524 0.841328i \(-0.318226\pi\)
0.540524 + 0.841328i \(0.318226\pi\)
\(824\) 6.33434e10 3.94418
\(825\) 1.30971e10 0.812059
\(826\) 1.23516e9 0.0762595
\(827\) −1.68089e10 −1.03341 −0.516703 0.856165i \(-0.672841\pi\)
−0.516703 + 0.856165i \(0.672841\pi\)
\(828\) 2.13509e9 0.130710
\(829\) 3.99422e9 0.243495 0.121748 0.992561i \(-0.461150\pi\)
0.121748 + 0.992561i \(0.461150\pi\)
\(830\) 5.31362e10 3.22565
\(831\) 7.95502e9 0.480881
\(832\) −1.16142e10 −0.699130
\(833\) 1.40847e9 0.0844287
\(834\) −3.09899e10 −1.84986
\(835\) −4.85121e10 −2.88368
\(836\) −6.98892e9 −0.413702
\(837\) 5.35965e9 0.315935
\(838\) 5.05435e10 2.96696
\(839\) −1.32657e10 −0.775467 −0.387734 0.921771i \(-0.626742\pi\)
−0.387734 + 0.921771i \(0.626742\pi\)
\(840\) −1.39424e10 −0.811632
\(841\) 8.37255e9 0.485369
\(842\) 1.16466e10 0.672367
\(843\) −1.66303e10 −0.956099
\(844\) −1.48627e10 −0.850938
\(845\) 1.90317e10 1.08512
\(846\) 7.29149e9 0.414019
\(847\) −2.07872e9 −0.117545
\(848\) −6.17372e10 −3.47665
\(849\) 1.21192e8 0.00679667
\(850\) 5.52899e9 0.308801
\(851\) −3.31333e8 −0.0184294
\(852\) −2.67255e10 −1.48043
\(853\) 3.06500e10 1.69087 0.845433 0.534082i \(-0.179343\pi\)
0.845433 + 0.534082i \(0.179343\pi\)
\(854\) 6.30015e9 0.346137
\(855\) −2.17115e9 −0.118798
\(856\) 5.54312e10 3.02062
\(857\) −5.05197e9 −0.274175 −0.137087 0.990559i \(-0.543774\pi\)
−0.137087 + 0.990559i \(0.543774\pi\)
\(858\) 9.26926e9 0.501002
\(859\) −6.74004e7 −0.00362816 −0.00181408 0.999998i \(-0.500577\pi\)
−0.00181408 + 0.999998i \(0.500577\pi\)
\(860\) 1.03712e11 5.56010
\(861\) 5.63154e9 0.300688
\(862\) 4.59703e10 2.44457
\(863\) 3.32493e10 1.76094 0.880471 0.474101i \(-0.157227\pi\)
0.880471 + 0.474101i \(0.157227\pi\)
\(864\) 7.27804e9 0.383898
\(865\) −4.47322e10 −2.34998
\(866\) −5.38627e9 −0.281822
\(867\) 1.09817e10 0.572273
\(868\) −2.43782e10 −1.26527
\(869\) 7.38623e9 0.381816
\(870\) 4.22318e10 2.17431
\(871\) 1.19846e10 0.614552
\(872\) 2.57802e9 0.131668
\(873\) 7.29192e9 0.370930
\(874\) −1.25840e9 −0.0637573
\(875\) −8.08137e9 −0.407809
\(876\) −1.37454e10 −0.690864
\(877\) −3.04224e10 −1.52298 −0.761491 0.648176i \(-0.775532\pi\)
−0.761491 + 0.648176i \(0.775532\pi\)
\(878\) −4.45616e10 −2.22193
\(879\) −9.64893e9 −0.479202
\(880\) 6.71365e10 3.32101
\(881\) 1.67047e10 0.823044 0.411522 0.911400i \(-0.364997\pi\)
0.411522 + 0.911400i \(0.364997\pi\)
\(882\) 1.13426e10 0.556639
\(883\) 4.43529e9 0.216800 0.108400 0.994107i \(-0.465427\pi\)
0.108400 + 0.994107i \(0.465427\pi\)
\(884\) 2.77574e9 0.135144
\(885\) −2.58203e9 −0.125216
\(886\) −7.35658e10 −3.55351
\(887\) 3.65741e10 1.75971 0.879855 0.475242i \(-0.157640\pi\)
0.879855 + 0.475242i \(0.157640\pi\)
\(888\) −3.69266e9 −0.176968
\(889\) 2.23101e9 0.106499
\(890\) −2.22308e10 −1.05704
\(891\) −1.85880e9 −0.0880360
\(892\) 3.56270e10 1.68075
\(893\) −3.04849e9 −0.143253
\(894\) 2.10435e10 0.985002
\(895\) 3.97958e10 1.85548
\(896\) −1.37041e9 −0.0636463
\(897\) 1.18391e9 0.0547703
\(898\) −3.03070e10 −1.39661
\(899\) 4.35869e10 2.00077
\(900\) 3.15846e10 1.44420
\(901\) 2.84507e9 0.129585
\(902\) −5.34221e10 −2.42381
\(903\) 5.51673e9 0.249330
\(904\) −9.20955e10 −4.14619
\(905\) −1.92794e10 −0.864618
\(906\) 7.73435e9 0.345522
\(907\) 1.91299e10 0.851307 0.425654 0.904886i \(-0.360044\pi\)
0.425654 + 0.904886i \(0.360044\pi\)
\(908\) 1.71616e10 0.760775
\(909\) −2.97718e9 −0.131471
\(910\) −1.30975e10 −0.576162
\(911\) −3.39426e10 −1.48741 −0.743704 0.668509i \(-0.766933\pi\)
−0.743704 + 0.668509i \(0.766933\pi\)
\(912\) −7.11908e9 −0.310772
\(913\) −1.90196e10 −0.827093
\(914\) 7.45422e8 0.0322917
\(915\) −1.31701e10 −0.568348
\(916\) −1.68649e10 −0.725018
\(917\) −7.69978e9 −0.329751
\(918\) −7.84696e8 −0.0334774
\(919\) −3.53345e10 −1.50174 −0.750870 0.660450i \(-0.770366\pi\)
−0.750870 + 0.660450i \(0.770366\pi\)
\(920\) 1.68929e10 0.715232
\(921\) 1.12669e10 0.475220
\(922\) 3.10111e10 1.30305
\(923\) −1.48193e10 −0.620330
\(924\) 8.45469e9 0.352570
\(925\) −4.90144e9 −0.203624
\(926\) 2.64332e10 1.09398
\(927\) −1.19328e10 −0.491997
\(928\) 5.91879e10 2.43117
\(929\) −3.23304e10 −1.32299 −0.661493 0.749951i \(-0.730077\pi\)
−0.661493 + 0.749951i \(0.730077\pi\)
\(930\) 7.18414e10 2.92876
\(931\) −4.74222e9 −0.192601
\(932\) −8.93354e10 −3.61467
\(933\) −4.24249e9 −0.171015
\(934\) −2.62854e10 −1.05560
\(935\) −3.09389e9 −0.123784
\(936\) 1.31945e10 0.525931
\(937\) 9.18986e9 0.364939 0.182469 0.983212i \(-0.441591\pi\)
0.182469 + 0.983212i \(0.441591\pi\)
\(938\) 1.54103e10 0.609679
\(939\) −2.30102e10 −0.906964
\(940\) 6.93296e10 2.72252
\(941\) −1.09465e10 −0.428265 −0.214133 0.976805i \(-0.568693\pi\)
−0.214133 + 0.976805i \(0.568693\pi\)
\(942\) 2.13962e10 0.833987
\(943\) −6.82331e9 −0.264974
\(944\) −8.46635e9 −0.327562
\(945\) 2.62650e9 0.101243
\(946\) −5.23329e10 −2.00982
\(947\) −2.72786e10 −1.04375 −0.521875 0.853022i \(-0.674767\pi\)
−0.521875 + 0.853022i \(0.674767\pi\)
\(948\) 1.78124e10 0.679036
\(949\) −7.62185e9 −0.289487
\(950\) −1.86157e10 −0.704445
\(951\) 1.99900e10 0.753670
\(952\) 2.10677e9 0.0791386
\(953\) −2.27671e10 −0.852085 −0.426043 0.904703i \(-0.640093\pi\)
−0.426043 + 0.904703i \(0.640093\pi\)
\(954\) 2.29118e10 0.854357
\(955\) 3.72665e10 1.38455
\(956\) −2.58452e10 −0.956704
\(957\) −1.51165e10 −0.557519
\(958\) −6.89227e10 −2.53270
\(959\) −1.43989e10 −0.527187
\(960\) 3.12188e10 1.13885
\(961\) 4.66339e10 1.69500
\(962\) −3.46891e9 −0.125626
\(963\) −1.04423e10 −0.376793
\(964\) −6.73298e10 −2.42068
\(965\) −6.34129e10 −2.27160
\(966\) 1.52233e9 0.0543360
\(967\) −1.38243e10 −0.491645 −0.245822 0.969315i \(-0.579058\pi\)
−0.245822 + 0.969315i \(0.579058\pi\)
\(968\) 2.80697e10 0.994660
\(969\) 3.28073e8 0.0115834
\(970\) 9.77418e10 3.43858
\(971\) 1.62886e10 0.570973 0.285487 0.958383i \(-0.407845\pi\)
0.285487 + 0.958383i \(0.407845\pi\)
\(972\) −4.48261e9 −0.156567
\(973\) −1.56739e10 −0.545483
\(974\) −3.66909e9 −0.127234
\(975\) 1.75137e10 0.605149
\(976\) −4.31840e10 −1.48679
\(977\) 1.93224e10 0.662874 0.331437 0.943477i \(-0.392467\pi\)
0.331437 + 0.943477i \(0.392467\pi\)
\(978\) 2.79074e10 0.953968
\(979\) 7.95733e9 0.271037
\(980\) 1.07849e11 3.66036
\(981\) −4.85654e8 −0.0164243
\(982\) 2.22021e10 0.748177
\(983\) 3.13862e10 1.05391 0.526953 0.849895i \(-0.323334\pi\)
0.526953 + 0.849895i \(0.323334\pi\)
\(984\) −7.60450e10 −2.54441
\(985\) 6.36735e10 2.12291
\(986\) −6.38146e9 −0.212007
\(987\) 3.68785e9 0.122085
\(988\) −9.34571e9 −0.308292
\(989\) −6.68420e9 −0.219716
\(990\) −2.49156e10 −0.816107
\(991\) −2.26508e9 −0.0739308 −0.0369654 0.999317i \(-0.511769\pi\)
−0.0369654 + 0.999317i \(0.511769\pi\)
\(992\) 1.00686e11 3.27475
\(993\) 5.84469e9 0.189426
\(994\) −1.90554e10 −0.615411
\(995\) 1.99430e10 0.641814
\(996\) −4.58670e10 −1.47093
\(997\) 4.99508e10 1.59628 0.798140 0.602472i \(-0.205817\pi\)
0.798140 + 0.602472i \(0.205817\pi\)
\(998\) 1.51273e10 0.481732
\(999\) 6.95633e8 0.0220750
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.1.2 17
3.2 odd 2 531.8.a.c.1.16 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.2 17 1.1 even 1 trivial
531.8.a.c.1.16 17 3.2 odd 2