Properties

Label 177.4.a.c.1.6
Level $177$
Weight $4$
Character 177.1
Self dual yes
Analytic conductor $10.443$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 49 x^{6} + 89 x^{5} + 648 x^{4} - 1023 x^{3} - 1476 x^{2} + 1940 x - 384\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.17127\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.17127 q^{2} -3.00000 q^{3} -3.28558 q^{4} +9.58086 q^{5} -6.51382 q^{6} +14.1591 q^{7} -24.5041 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.17127 q^{2} -3.00000 q^{3} -3.28558 q^{4} +9.58086 q^{5} -6.51382 q^{6} +14.1591 q^{7} -24.5041 q^{8} +9.00000 q^{9} +20.8027 q^{10} +19.1455 q^{11} +9.85673 q^{12} +15.5364 q^{13} +30.7433 q^{14} -28.7426 q^{15} -26.9204 q^{16} +100.737 q^{17} +19.5414 q^{18} +74.3408 q^{19} -31.4787 q^{20} -42.4773 q^{21} +41.5701 q^{22} +98.9130 q^{23} +73.5122 q^{24} -33.2071 q^{25} +33.7338 q^{26} -27.0000 q^{27} -46.5208 q^{28} +194.021 q^{29} -62.4080 q^{30} +52.9293 q^{31} +137.581 q^{32} -57.4366 q^{33} +218.728 q^{34} +135.656 q^{35} -29.5702 q^{36} -212.802 q^{37} +161.414 q^{38} -46.6093 q^{39} -234.770 q^{40} -395.318 q^{41} -92.2298 q^{42} +305.894 q^{43} -62.9041 q^{44} +86.2277 q^{45} +214.767 q^{46} -630.864 q^{47} +80.7611 q^{48} -142.520 q^{49} -72.1018 q^{50} -302.211 q^{51} -51.0461 q^{52} -109.277 q^{53} -58.6243 q^{54} +183.431 q^{55} -346.956 q^{56} -223.022 q^{57} +421.272 q^{58} -59.0000 q^{59} +94.4360 q^{60} +240.030 q^{61} +114.924 q^{62} +127.432 q^{63} +514.089 q^{64} +148.852 q^{65} -124.710 q^{66} -100.162 q^{67} -330.980 q^{68} -296.739 q^{69} +294.547 q^{70} +263.939 q^{71} -220.537 q^{72} -296.815 q^{73} -462.051 q^{74} +99.6214 q^{75} -244.253 q^{76} +271.083 q^{77} -101.201 q^{78} +626.260 q^{79} -257.920 q^{80} +81.0000 q^{81} -858.343 q^{82} -7.08512 q^{83} +139.562 q^{84} +965.148 q^{85} +664.179 q^{86} -582.062 q^{87} -469.143 q^{88} -132.516 q^{89} +187.224 q^{90} +219.982 q^{91} -324.986 q^{92} -158.788 q^{93} -1369.78 q^{94} +712.249 q^{95} -412.743 q^{96} +1476.73 q^{97} -309.449 q^{98} +172.310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{2} - 24q^{3} + 38q^{4} - 12q^{5} - 6q^{6} + 53q^{7} + 3q^{8} + 72q^{9} + O(q^{10}) \) \( 8q + 2q^{2} - 24q^{3} + 38q^{4} - 12q^{5} - 6q^{6} + 53q^{7} + 3q^{8} + 72q^{9} + 29q^{10} - 27q^{11} - 114q^{12} + 89q^{13} - 37q^{14} + 36q^{15} + 362q^{16} + 79q^{17} + 18q^{18} + 288q^{19} + 457q^{20} - 159q^{21} + 596q^{22} + 202q^{23} - 9q^{24} + 264q^{25} + 270q^{26} - 216q^{27} + 702q^{28} - 114q^{29} - 87q^{30} + 538q^{31} + 316q^{32} + 81q^{33} + 498q^{34} - 196q^{35} + 342q^{36} + 395q^{37} + 397q^{38} - 267q^{39} + 918q^{40} - 39q^{41} + 111q^{42} + 527q^{43} + 64q^{44} - 108q^{45} - 539q^{46} + 860q^{47} - 1086q^{48} + 347q^{49} - 591q^{50} - 237q^{51} - 644q^{52} - 812q^{53} - 54q^{54} + 536q^{55} - 2218q^{56} - 864q^{57} - 1154q^{58} - 472q^{59} - 1371q^{60} - 460q^{61} - 2014q^{62} + 477q^{63} - 451q^{64} - 986q^{65} - 1788q^{66} + 1934q^{67} - 69q^{68} - 606q^{69} - 1028q^{70} - 1687q^{71} + 27q^{72} + 1980q^{73} - 2400q^{74} - 792q^{75} - 940q^{76} - 821q^{77} - 810q^{78} + 3319q^{79} - 2119q^{80} + 648q^{81} + 429q^{82} + 2057q^{83} - 2106q^{84} + 566q^{85} - 6690q^{86} + 342q^{87} + 1189q^{88} + 1668q^{89} + 261q^{90} + 2427q^{91} - 980q^{92} - 1614q^{93} + 332q^{94} + 2146q^{95} - 948q^{96} + 1956q^{97} - 2026q^{98} - 243q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.17127 0.767661 0.383830 0.923404i \(-0.374605\pi\)
0.383830 + 0.923404i \(0.374605\pi\)
\(3\) −3.00000 −0.577350
\(4\) −3.28558 −0.410697
\(5\) 9.58086 0.856938 0.428469 0.903556i \(-0.359053\pi\)
0.428469 + 0.903556i \(0.359053\pi\)
\(6\) −6.51382 −0.443209
\(7\) 14.1591 0.764520 0.382260 0.924055i \(-0.375146\pi\)
0.382260 + 0.924055i \(0.375146\pi\)
\(8\) −24.5041 −1.08294
\(9\) 9.00000 0.333333
\(10\) 20.8027 0.657838
\(11\) 19.1455 0.524781 0.262391 0.964962i \(-0.415489\pi\)
0.262391 + 0.964962i \(0.415489\pi\)
\(12\) 9.85673 0.237116
\(13\) 15.5364 0.331464 0.165732 0.986171i \(-0.447001\pi\)
0.165732 + 0.986171i \(0.447001\pi\)
\(14\) 30.7433 0.586892
\(15\) −28.7426 −0.494753
\(16\) −26.9204 −0.420631
\(17\) 100.737 1.43720 0.718598 0.695426i \(-0.244784\pi\)
0.718598 + 0.695426i \(0.244784\pi\)
\(18\) 19.5414 0.255887
\(19\) 74.3408 0.897629 0.448815 0.893625i \(-0.351846\pi\)
0.448815 + 0.893625i \(0.351846\pi\)
\(20\) −31.4787 −0.351942
\(21\) −42.4773 −0.441396
\(22\) 41.5701 0.402854
\(23\) 98.9130 0.896730 0.448365 0.893851i \(-0.352006\pi\)
0.448365 + 0.893851i \(0.352006\pi\)
\(24\) 73.5122 0.625234
\(25\) −33.2071 −0.265657
\(26\) 33.7338 0.254452
\(27\) −27.0000 −0.192450
\(28\) −46.5208 −0.313986
\(29\) 194.021 1.24237 0.621185 0.783664i \(-0.286652\pi\)
0.621185 + 0.783664i \(0.286652\pi\)
\(30\) −62.4080 −0.379803
\(31\) 52.9293 0.306658 0.153329 0.988175i \(-0.451001\pi\)
0.153329 + 0.988175i \(0.451001\pi\)
\(32\) 137.581 0.760035
\(33\) −57.4366 −0.302982
\(34\) 218.728 1.10328
\(35\) 135.656 0.655146
\(36\) −29.5702 −0.136899
\(37\) −212.802 −0.945525 −0.472762 0.881190i \(-0.656743\pi\)
−0.472762 + 0.881190i \(0.656743\pi\)
\(38\) 161.414 0.689075
\(39\) −46.6093 −0.191371
\(40\) −234.770 −0.928010
\(41\) −395.318 −1.50581 −0.752906 0.658128i \(-0.771348\pi\)
−0.752906 + 0.658128i \(0.771348\pi\)
\(42\) −92.2298 −0.338842
\(43\) 305.894 1.08485 0.542423 0.840106i \(-0.317507\pi\)
0.542423 + 0.840106i \(0.317507\pi\)
\(44\) −62.9041 −0.215526
\(45\) 86.2277 0.285646
\(46\) 214.767 0.688384
\(47\) −630.864 −1.95789 −0.978947 0.204116i \(-0.934568\pi\)
−0.978947 + 0.204116i \(0.934568\pi\)
\(48\) 80.7611 0.242851
\(49\) −142.520 −0.415510
\(50\) −72.1018 −0.203935
\(51\) −302.211 −0.829766
\(52\) −51.0461 −0.136131
\(53\) −109.277 −0.283214 −0.141607 0.989923i \(-0.545227\pi\)
−0.141607 + 0.989923i \(0.545227\pi\)
\(54\) −58.6243 −0.147736
\(55\) 183.431 0.449705
\(56\) −346.956 −0.827926
\(57\) −223.022 −0.518246
\(58\) 421.272 0.953718
\(59\) −59.0000 −0.130189
\(60\) 94.4360 0.203194
\(61\) 240.030 0.503815 0.251907 0.967751i \(-0.418942\pi\)
0.251907 + 0.967751i \(0.418942\pi\)
\(62\) 114.924 0.235409
\(63\) 127.432 0.254840
\(64\) 514.089 1.00408
\(65\) 148.852 0.284044
\(66\) −124.710 −0.232588
\(67\) −100.162 −0.182638 −0.0913191 0.995822i \(-0.529108\pi\)
−0.0913191 + 0.995822i \(0.529108\pi\)
\(68\) −330.980 −0.590252
\(69\) −296.739 −0.517727
\(70\) 294.547 0.502930
\(71\) 263.939 0.441180 0.220590 0.975367i \(-0.429202\pi\)
0.220590 + 0.975367i \(0.429202\pi\)
\(72\) −220.537 −0.360979
\(73\) −296.815 −0.475884 −0.237942 0.971279i \(-0.576473\pi\)
−0.237942 + 0.971279i \(0.576473\pi\)
\(74\) −462.051 −0.725842
\(75\) 99.6214 0.153377
\(76\) −244.253 −0.368654
\(77\) 271.083 0.401205
\(78\) −101.201 −0.146908
\(79\) 626.260 0.891896 0.445948 0.895059i \(-0.352867\pi\)
0.445948 + 0.895059i \(0.352867\pi\)
\(80\) −257.920 −0.360454
\(81\) 81.0000 0.111111
\(82\) −858.343 −1.15595
\(83\) −7.08512 −0.00936980 −0.00468490 0.999989i \(-0.501491\pi\)
−0.00468490 + 0.999989i \(0.501491\pi\)
\(84\) 139.562 0.181280
\(85\) 965.148 1.23159
\(86\) 664.179 0.832793
\(87\) −582.062 −0.717283
\(88\) −469.143 −0.568305
\(89\) −132.516 −0.157828 −0.0789138 0.996881i \(-0.525145\pi\)
−0.0789138 + 0.996881i \(0.525145\pi\)
\(90\) 187.224 0.219279
\(91\) 219.982 0.253411
\(92\) −324.986 −0.368284
\(93\) −158.788 −0.177049
\(94\) −1369.78 −1.50300
\(95\) 712.249 0.769213
\(96\) −412.743 −0.438806
\(97\) 1476.73 1.54577 0.772885 0.634546i \(-0.218813\pi\)
0.772885 + 0.634546i \(0.218813\pi\)
\(98\) −309.449 −0.318970
\(99\) 172.310 0.174927
\(100\) 109.105 0.109105
\(101\) −1349.75 −1.32975 −0.664875 0.746955i \(-0.731515\pi\)
−0.664875 + 0.746955i \(0.731515\pi\)
\(102\) −656.183 −0.636978
\(103\) 1258.15 1.20359 0.601794 0.798651i \(-0.294453\pi\)
0.601794 + 0.798651i \(0.294453\pi\)
\(104\) −380.705 −0.358954
\(105\) −406.969 −0.378249
\(106\) −237.270 −0.217412
\(107\) −1690.18 −1.52706 −0.763532 0.645771i \(-0.776536\pi\)
−0.763532 + 0.645771i \(0.776536\pi\)
\(108\) 88.7106 0.0790387
\(109\) −1293.50 −1.13665 −0.568324 0.822805i \(-0.692408\pi\)
−0.568324 + 0.822805i \(0.692408\pi\)
\(110\) 398.278 0.345221
\(111\) 638.406 0.545899
\(112\) −381.168 −0.321580
\(113\) 1086.11 0.904185 0.452093 0.891971i \(-0.350678\pi\)
0.452093 + 0.891971i \(0.350678\pi\)
\(114\) −484.243 −0.397837
\(115\) 947.672 0.768442
\(116\) −637.470 −0.510238
\(117\) 139.828 0.110488
\(118\) −128.105 −0.0999409
\(119\) 1426.35 1.09876
\(120\) 704.310 0.535787
\(121\) −964.449 −0.724605
\(122\) 521.171 0.386759
\(123\) 1185.95 0.869381
\(124\) −173.903 −0.125943
\(125\) −1515.76 −1.08459
\(126\) 276.689 0.195631
\(127\) −681.878 −0.476432 −0.238216 0.971212i \(-0.576563\pi\)
−0.238216 + 0.971212i \(0.576563\pi\)
\(128\) 15.5784 0.0107574
\(129\) −917.681 −0.626336
\(130\) 323.199 0.218049
\(131\) −1373.82 −0.916267 −0.458134 0.888883i \(-0.651482\pi\)
−0.458134 + 0.888883i \(0.651482\pi\)
\(132\) 188.712 0.124434
\(133\) 1052.60 0.686255
\(134\) −217.479 −0.140204
\(135\) −258.683 −0.164918
\(136\) −2468.47 −1.55639
\(137\) −920.396 −0.573976 −0.286988 0.957934i \(-0.592654\pi\)
−0.286988 + 0.957934i \(0.592654\pi\)
\(138\) −644.301 −0.397439
\(139\) 2435.03 1.48587 0.742936 0.669362i \(-0.233433\pi\)
0.742936 + 0.669362i \(0.233433\pi\)
\(140\) −445.709 −0.269067
\(141\) 1892.59 1.13039
\(142\) 573.083 0.338676
\(143\) 297.453 0.173946
\(144\) −242.283 −0.140210
\(145\) 1858.88 1.06463
\(146\) −644.465 −0.365317
\(147\) 427.559 0.239895
\(148\) 699.177 0.388324
\(149\) 1240.27 0.681925 0.340963 0.940077i \(-0.389247\pi\)
0.340963 + 0.940077i \(0.389247\pi\)
\(150\) 216.305 0.117742
\(151\) 3384.25 1.82389 0.911943 0.410318i \(-0.134582\pi\)
0.911943 + 0.410318i \(0.134582\pi\)
\(152\) −1821.65 −0.972076
\(153\) 906.634 0.479065
\(154\) 588.596 0.307990
\(155\) 507.108 0.262787
\(156\) 153.138 0.0785954
\(157\) 195.675 0.0994685 0.0497342 0.998762i \(-0.484163\pi\)
0.0497342 + 0.998762i \(0.484163\pi\)
\(158\) 1359.78 0.684674
\(159\) 327.830 0.163514
\(160\) 1318.14 0.651303
\(161\) 1400.52 0.685568
\(162\) 175.873 0.0852956
\(163\) 135.992 0.0653479 0.0326739 0.999466i \(-0.489598\pi\)
0.0326739 + 0.999466i \(0.489598\pi\)
\(164\) 1298.85 0.618433
\(165\) −550.292 −0.259637
\(166\) −15.3837 −0.00719283
\(167\) −627.557 −0.290789 −0.145395 0.989374i \(-0.546445\pi\)
−0.145395 + 0.989374i \(0.546445\pi\)
\(168\) 1040.87 0.478004
\(169\) −1955.62 −0.890132
\(170\) 2095.60 0.945442
\(171\) 669.067 0.299210
\(172\) −1005.04 −0.445543
\(173\) −2869.43 −1.26103 −0.630515 0.776177i \(-0.717156\pi\)
−0.630515 + 0.776177i \(0.717156\pi\)
\(174\) −1263.81 −0.550630
\(175\) −470.183 −0.203100
\(176\) −515.404 −0.220739
\(177\) 177.000 0.0751646
\(178\) −287.728 −0.121158
\(179\) 2266.38 0.946353 0.473176 0.880968i \(-0.343107\pi\)
0.473176 + 0.880968i \(0.343107\pi\)
\(180\) −283.308 −0.117314
\(181\) 449.089 0.184423 0.0922114 0.995739i \(-0.470606\pi\)
0.0922114 + 0.995739i \(0.470606\pi\)
\(182\) 477.640 0.194533
\(183\) −720.090 −0.290878
\(184\) −2423.77 −0.971102
\(185\) −2038.82 −0.810256
\(186\) −344.772 −0.135913
\(187\) 1928.66 0.754213
\(188\) 2072.75 0.804101
\(189\) −382.296 −0.147132
\(190\) 1546.49 0.590494
\(191\) 495.207 0.187602 0.0938009 0.995591i \(-0.470098\pi\)
0.0938009 + 0.995591i \(0.470098\pi\)
\(192\) −1542.27 −0.579706
\(193\) −2118.19 −0.790002 −0.395001 0.918681i \(-0.629256\pi\)
−0.395001 + 0.918681i \(0.629256\pi\)
\(194\) 3206.39 1.18663
\(195\) −446.557 −0.163993
\(196\) 468.260 0.170649
\(197\) 372.268 0.134635 0.0673173 0.997732i \(-0.478556\pi\)
0.0673173 + 0.997732i \(0.478556\pi\)
\(198\) 374.131 0.134285
\(199\) 989.744 0.352568 0.176284 0.984339i \(-0.443592\pi\)
0.176284 + 0.984339i \(0.443592\pi\)
\(200\) 813.710 0.287690
\(201\) 300.487 0.105446
\(202\) −2930.66 −1.02080
\(203\) 2747.16 0.949816
\(204\) 992.939 0.340782
\(205\) −3787.48 −1.29039
\(206\) 2731.80 0.923948
\(207\) 890.217 0.298910
\(208\) −418.246 −0.139424
\(209\) 1423.29 0.471059
\(210\) −883.641 −0.290367
\(211\) −1268.28 −0.413802 −0.206901 0.978362i \(-0.566338\pi\)
−0.206901 + 0.978362i \(0.566338\pi\)
\(212\) 359.037 0.116315
\(213\) −791.816 −0.254715
\(214\) −3669.84 −1.17227
\(215\) 2930.72 0.929645
\(216\) 661.610 0.208411
\(217\) 749.432 0.234446
\(218\) −2808.54 −0.872560
\(219\) 890.444 0.274752
\(220\) −602.675 −0.184692
\(221\) 1565.09 0.476378
\(222\) 1386.15 0.419065
\(223\) 1129.78 0.339262 0.169631 0.985508i \(-0.445742\pi\)
0.169631 + 0.985508i \(0.445742\pi\)
\(224\) 1948.02 0.581062
\(225\) −298.864 −0.0885524
\(226\) 2358.25 0.694108
\(227\) −161.210 −0.0471359 −0.0235680 0.999722i \(-0.507503\pi\)
−0.0235680 + 0.999722i \(0.507503\pi\)
\(228\) 732.758 0.212842
\(229\) 118.980 0.0343338 0.0171669 0.999853i \(-0.494535\pi\)
0.0171669 + 0.999853i \(0.494535\pi\)
\(230\) 2057.65 0.589903
\(231\) −813.250 −0.231636
\(232\) −4754.29 −1.34541
\(233\) −4469.51 −1.25668 −0.628342 0.777937i \(-0.716266\pi\)
−0.628342 + 0.777937i \(0.716266\pi\)
\(234\) 303.604 0.0848172
\(235\) −6044.22 −1.67779
\(236\) 193.849 0.0534682
\(237\) −1878.78 −0.514937
\(238\) 3096.99 0.843479
\(239\) −4682.67 −1.26735 −0.633675 0.773600i \(-0.718454\pi\)
−0.633675 + 0.773600i \(0.718454\pi\)
\(240\) 773.761 0.208108
\(241\) −3836.61 −1.02547 −0.512735 0.858547i \(-0.671367\pi\)
−0.512735 + 0.858547i \(0.671367\pi\)
\(242\) −2094.08 −0.556251
\(243\) −243.000 −0.0641500
\(244\) −788.638 −0.206915
\(245\) −1365.46 −0.356066
\(246\) 2575.03 0.667389
\(247\) 1154.99 0.297531
\(248\) −1296.98 −0.332091
\(249\) 21.2554 0.00540966
\(250\) −3291.13 −0.832597
\(251\) −331.121 −0.0832677 −0.0416338 0.999133i \(-0.513256\pi\)
−0.0416338 + 0.999133i \(0.513256\pi\)
\(252\) −418.687 −0.104662
\(253\) 1893.74 0.470587
\(254\) −1480.54 −0.365738
\(255\) −2895.44 −0.711058
\(256\) −4078.89 −0.995822
\(257\) 1423.82 0.345585 0.172792 0.984958i \(-0.444721\pi\)
0.172792 + 0.984958i \(0.444721\pi\)
\(258\) −1992.54 −0.480813
\(259\) −3013.08 −0.722872
\(260\) −489.066 −0.116656
\(261\) 1746.19 0.414123
\(262\) −2982.93 −0.703382
\(263\) −5205.39 −1.22045 −0.610225 0.792228i \(-0.708921\pi\)
−0.610225 + 0.792228i \(0.708921\pi\)
\(264\) 1407.43 0.328111
\(265\) −1046.97 −0.242697
\(266\) 2285.48 0.526811
\(267\) 397.548 0.0911218
\(268\) 329.091 0.0750090
\(269\) 349.224 0.0791544 0.0395772 0.999217i \(-0.487399\pi\)
0.0395772 + 0.999217i \(0.487399\pi\)
\(270\) −561.672 −0.126601
\(271\) −8231.66 −1.84516 −0.922579 0.385808i \(-0.873923\pi\)
−0.922579 + 0.385808i \(0.873923\pi\)
\(272\) −2711.88 −0.604529
\(273\) −659.945 −0.146307
\(274\) −1998.43 −0.440619
\(275\) −635.768 −0.139412
\(276\) 974.959 0.212629
\(277\) −7242.28 −1.57093 −0.785463 0.618908i \(-0.787575\pi\)
−0.785463 + 0.618908i \(0.787575\pi\)
\(278\) 5287.11 1.14065
\(279\) 476.364 0.102219
\(280\) −3324.13 −0.709482
\(281\) 1769.76 0.375713 0.187856 0.982197i \(-0.439846\pi\)
0.187856 + 0.982197i \(0.439846\pi\)
\(282\) 4109.33 0.867756
\(283\) 5303.35 1.11396 0.556981 0.830525i \(-0.311960\pi\)
0.556981 + 0.830525i \(0.311960\pi\)
\(284\) −867.191 −0.181191
\(285\) −2136.75 −0.444105
\(286\) 645.851 0.133531
\(287\) −5597.35 −1.15122
\(288\) 1238.23 0.253345
\(289\) 5234.96 1.06553
\(290\) 4036.14 0.817278
\(291\) −4430.20 −0.892451
\(292\) 975.207 0.195444
\(293\) 138.305 0.0275764 0.0137882 0.999905i \(-0.495611\pi\)
0.0137882 + 0.999905i \(0.495611\pi\)
\(294\) 928.348 0.184158
\(295\) −565.271 −0.111564
\(296\) 5214.51 1.02394
\(297\) −516.929 −0.100994
\(298\) 2692.96 0.523487
\(299\) 1536.75 0.297233
\(300\) −327.314 −0.0629916
\(301\) 4331.18 0.829386
\(302\) 7348.14 1.40012
\(303\) 4049.24 0.767731
\(304\) −2001.28 −0.377570
\(305\) 2299.70 0.431738
\(306\) 1968.55 0.367760
\(307\) 2464.57 0.458178 0.229089 0.973406i \(-0.426425\pi\)
0.229089 + 0.973406i \(0.426425\pi\)
\(308\) −890.665 −0.164774
\(309\) −3774.46 −0.694892
\(310\) 1101.07 0.201731
\(311\) −9275.20 −1.69115 −0.845576 0.533855i \(-0.820743\pi\)
−0.845576 + 0.533855i \(0.820743\pi\)
\(312\) 1142.12 0.207242
\(313\) 5699.12 1.02918 0.514590 0.857437i \(-0.327944\pi\)
0.514590 + 0.857437i \(0.327944\pi\)
\(314\) 424.863 0.0763580
\(315\) 1220.91 0.218382
\(316\) −2057.63 −0.366299
\(317\) −859.960 −0.152366 −0.0761832 0.997094i \(-0.524273\pi\)
−0.0761832 + 0.997094i \(0.524273\pi\)
\(318\) 711.809 0.125523
\(319\) 3714.63 0.651972
\(320\) 4925.41 0.860434
\(321\) 5070.54 0.881650
\(322\) 3040.91 0.526283
\(323\) 7488.88 1.29007
\(324\) −266.132 −0.0456330
\(325\) −515.920 −0.0880557
\(326\) 295.275 0.0501650
\(327\) 3880.49 0.656244
\(328\) 9686.89 1.63070
\(329\) −8932.47 −1.49685
\(330\) −1194.83 −0.199313
\(331\) 10291.1 1.70891 0.854454 0.519527i \(-0.173892\pi\)
0.854454 + 0.519527i \(0.173892\pi\)
\(332\) 23.2787 0.00384815
\(333\) −1915.22 −0.315175
\(334\) −1362.60 −0.223227
\(335\) −959.640 −0.156510
\(336\) 1143.50 0.185665
\(337\) 1861.32 0.300869 0.150434 0.988620i \(-0.451933\pi\)
0.150434 + 0.988620i \(0.451933\pi\)
\(338\) −4246.18 −0.683319
\(339\) −3258.34 −0.522032
\(340\) −3171.07 −0.505810
\(341\) 1013.36 0.160928
\(342\) 1452.73 0.229692
\(343\) −6874.52 −1.08218
\(344\) −7495.64 −1.17482
\(345\) −2843.01 −0.443660
\(346\) −6230.30 −0.968044
\(347\) −2342.82 −0.362447 −0.181224 0.983442i \(-0.558006\pi\)
−0.181224 + 0.983442i \(0.558006\pi\)
\(348\) 1912.41 0.294586
\(349\) −9507.00 −1.45816 −0.729081 0.684428i \(-0.760052\pi\)
−0.729081 + 0.684428i \(0.760052\pi\)
\(350\) −1020.90 −0.155912
\(351\) −419.483 −0.0637902
\(352\) 2634.06 0.398852
\(353\) 7366.56 1.11071 0.555357 0.831612i \(-0.312581\pi\)
0.555357 + 0.831612i \(0.312581\pi\)
\(354\) 384.315 0.0577009
\(355\) 2528.76 0.378064
\(356\) 435.391 0.0648193
\(357\) −4279.04 −0.634372
\(358\) 4920.93 0.726478
\(359\) 7674.15 1.12821 0.564103 0.825704i \(-0.309222\pi\)
0.564103 + 0.825704i \(0.309222\pi\)
\(360\) −2112.93 −0.309337
\(361\) −1332.44 −0.194262
\(362\) 975.095 0.141574
\(363\) 2893.35 0.418351
\(364\) −722.767 −0.104075
\(365\) −2843.74 −0.407803
\(366\) −1563.51 −0.223295
\(367\) 4845.83 0.689238 0.344619 0.938743i \(-0.388008\pi\)
0.344619 + 0.938743i \(0.388008\pi\)
\(368\) −2662.77 −0.377192
\(369\) −3557.86 −0.501937
\(370\) −4426.84 −0.622002
\(371\) −1547.26 −0.216522
\(372\) 521.710 0.0727135
\(373\) −3677.62 −0.510510 −0.255255 0.966874i \(-0.582159\pi\)
−0.255255 + 0.966874i \(0.582159\pi\)
\(374\) 4187.65 0.578980
\(375\) 4547.28 0.626188
\(376\) 15458.7 2.12027
\(377\) 3014.39 0.411800
\(378\) −830.068 −0.112947
\(379\) 2448.51 0.331851 0.165925 0.986138i \(-0.446939\pi\)
0.165925 + 0.986138i \(0.446939\pi\)
\(380\) −2340.15 −0.315913
\(381\) 2045.63 0.275068
\(382\) 1075.23 0.144014
\(383\) 7777.55 1.03764 0.518818 0.854885i \(-0.326372\pi\)
0.518818 + 0.854885i \(0.326372\pi\)
\(384\) −46.7352 −0.00621080
\(385\) 2597.21 0.343808
\(386\) −4599.16 −0.606453
\(387\) 2753.04 0.361615
\(388\) −4851.93 −0.634843
\(389\) −13158.8 −1.71512 −0.857558 0.514387i \(-0.828019\pi\)
−0.857558 + 0.514387i \(0.828019\pi\)
\(390\) −969.596 −0.125891
\(391\) 9964.21 1.28878
\(392\) 3492.31 0.449971
\(393\) 4121.45 0.529007
\(394\) 808.296 0.103354
\(395\) 6000.11 0.764300
\(396\) −566.137 −0.0718420
\(397\) −7496.45 −0.947698 −0.473849 0.880606i \(-0.657136\pi\)
−0.473849 + 0.880606i \(0.657136\pi\)
\(398\) 2149.00 0.270653
\(399\) −3157.80 −0.396210
\(400\) 893.949 0.111744
\(401\) −4346.75 −0.541313 −0.270657 0.962676i \(-0.587241\pi\)
−0.270657 + 0.962676i \(0.587241\pi\)
\(402\) 652.438 0.0809469
\(403\) 822.332 0.101646
\(404\) 4434.69 0.546124
\(405\) 776.050 0.0952153
\(406\) 5964.83 0.729136
\(407\) −4074.20 −0.496193
\(408\) 7405.40 0.898584
\(409\) 14635.7 1.76941 0.884706 0.466149i \(-0.154359\pi\)
0.884706 + 0.466149i \(0.154359\pi\)
\(410\) −8223.66 −0.990580
\(411\) 2761.19 0.331385
\(412\) −4133.76 −0.494311
\(413\) −835.387 −0.0995320
\(414\) 1932.90 0.229461
\(415\) −67.8816 −0.00802934
\(416\) 2137.52 0.251924
\(417\) −7305.08 −0.857869
\(418\) 3090.36 0.361613
\(419\) −6292.05 −0.733620 −0.366810 0.930296i \(-0.619550\pi\)
−0.366810 + 0.930296i \(0.619550\pi\)
\(420\) 1337.13 0.155346
\(421\) −10784.3 −1.24845 −0.624223 0.781246i \(-0.714584\pi\)
−0.624223 + 0.781246i \(0.714584\pi\)
\(422\) −2753.79 −0.317660
\(423\) −5677.78 −0.652631
\(424\) 2677.73 0.306702
\(425\) −3345.19 −0.381802
\(426\) −1719.25 −0.195535
\(427\) 3398.61 0.385176
\(428\) 5553.21 0.627160
\(429\) −892.359 −0.100428
\(430\) 6363.40 0.713652
\(431\) 5954.37 0.665457 0.332728 0.943023i \(-0.392031\pi\)
0.332728 + 0.943023i \(0.392031\pi\)
\(432\) 726.850 0.0809504
\(433\) 4561.77 0.506292 0.253146 0.967428i \(-0.418535\pi\)
0.253146 + 0.967428i \(0.418535\pi\)
\(434\) 1627.22 0.179975
\(435\) −5576.65 −0.614667
\(436\) 4249.89 0.466818
\(437\) 7353.28 0.804931
\(438\) 1933.40 0.210916
\(439\) −7301.09 −0.793763 −0.396882 0.917870i \(-0.629908\pi\)
−0.396882 + 0.917870i \(0.629908\pi\)
\(440\) −4494.79 −0.487002
\(441\) −1282.68 −0.138503
\(442\) 3398.25 0.365697
\(443\) −15813.1 −1.69595 −0.847973 0.530040i \(-0.822177\pi\)
−0.847973 + 0.530040i \(0.822177\pi\)
\(444\) −2097.53 −0.224199
\(445\) −1269.62 −0.135248
\(446\) 2453.05 0.260438
\(447\) −3720.81 −0.393710
\(448\) 7279.04 0.767639
\(449\) 1604.95 0.168691 0.0843456 0.996437i \(-0.473120\pi\)
0.0843456 + 0.996437i \(0.473120\pi\)
\(450\) −648.916 −0.0679782
\(451\) −7568.57 −0.790221
\(452\) −3568.51 −0.371346
\(453\) −10152.8 −1.05302
\(454\) −350.030 −0.0361844
\(455\) 2107.61 0.217157
\(456\) 5464.96 0.561228
\(457\) 9914.58 1.01485 0.507423 0.861697i \(-0.330598\pi\)
0.507423 + 0.861697i \(0.330598\pi\)
\(458\) 258.339 0.0263567
\(459\) −2719.90 −0.276589
\(460\) −3113.65 −0.315597
\(461\) −5717.04 −0.577590 −0.288795 0.957391i \(-0.593255\pi\)
−0.288795 + 0.957391i \(0.593255\pi\)
\(462\) −1765.79 −0.177818
\(463\) 15831.2 1.58906 0.794532 0.607222i \(-0.207716\pi\)
0.794532 + 0.607222i \(0.207716\pi\)
\(464\) −5223.11 −0.522579
\(465\) −1521.33 −0.151720
\(466\) −9704.53 −0.964707
\(467\) 375.189 0.0371770 0.0185885 0.999827i \(-0.494083\pi\)
0.0185885 + 0.999827i \(0.494083\pi\)
\(468\) −459.415 −0.0453771
\(469\) −1418.21 −0.139630
\(470\) −13123.6 −1.28798
\(471\) −587.024 −0.0574281
\(472\) 1445.74 0.140986
\(473\) 5856.49 0.569306
\(474\) −4079.35 −0.395296
\(475\) −2468.65 −0.238462
\(476\) −4686.37 −0.451260
\(477\) −983.491 −0.0944046
\(478\) −10167.3 −0.972894
\(479\) −3601.59 −0.343551 −0.171775 0.985136i \(-0.554950\pi\)
−0.171775 + 0.985136i \(0.554950\pi\)
\(480\) −3954.43 −0.376030
\(481\) −3306.18 −0.313407
\(482\) −8330.33 −0.787212
\(483\) −4201.56 −0.395813
\(484\) 3168.77 0.297593
\(485\) 14148.4 1.32463
\(486\) −527.619 −0.0492455
\(487\) −14132.2 −1.31498 −0.657488 0.753465i \(-0.728381\pi\)
−0.657488 + 0.753465i \(0.728381\pi\)
\(488\) −5881.71 −0.545600
\(489\) −407.976 −0.0377286
\(490\) −2964.79 −0.273338
\(491\) −293.022 −0.0269326 −0.0134663 0.999909i \(-0.504287\pi\)
−0.0134663 + 0.999909i \(0.504287\pi\)
\(492\) −3896.54 −0.357052
\(493\) 19545.1 1.78553
\(494\) 2507.80 0.228403
\(495\) 1650.87 0.149902
\(496\) −1424.88 −0.128990
\(497\) 3737.14 0.337291
\(498\) 46.1512 0.00415278
\(499\) 15571.2 1.39692 0.698459 0.715650i \(-0.253869\pi\)
0.698459 + 0.715650i \(0.253869\pi\)
\(500\) 4980.15 0.445438
\(501\) 1882.67 0.167887
\(502\) −718.954 −0.0639213
\(503\) 12697.7 1.12557 0.562784 0.826604i \(-0.309730\pi\)
0.562784 + 0.826604i \(0.309730\pi\)
\(504\) −3122.60 −0.275975
\(505\) −12931.7 −1.13951
\(506\) 4111.83 0.361251
\(507\) 5866.86 0.513918
\(508\) 2240.36 0.195669
\(509\) −4294.84 −0.373999 −0.187000 0.982360i \(-0.559876\pi\)
−0.187000 + 0.982360i \(0.559876\pi\)
\(510\) −6286.80 −0.545851
\(511\) −4202.63 −0.363822
\(512\) −8981.00 −0.775210
\(513\) −2007.20 −0.172749
\(514\) 3091.49 0.265292
\(515\) 12054.2 1.03140
\(516\) 3015.11 0.257234
\(517\) −12078.2 −1.02747
\(518\) −6542.22 −0.554921
\(519\) 8608.28 0.728057
\(520\) −3647.48 −0.307601
\(521\) −22351.1 −1.87950 −0.939751 0.341860i \(-0.888943\pi\)
−0.939751 + 0.341860i \(0.888943\pi\)
\(522\) 3791.44 0.317906
\(523\) 12800.7 1.07024 0.535118 0.844777i \(-0.320267\pi\)
0.535118 + 0.844777i \(0.320267\pi\)
\(524\) 4513.78 0.376308
\(525\) 1410.55 0.117260
\(526\) −11302.3 −0.936891
\(527\) 5331.95 0.440727
\(528\) 1546.21 0.127444
\(529\) −2383.22 −0.195875
\(530\) −2273.25 −0.186309
\(531\) −531.000 −0.0433963
\(532\) −3458.40 −0.281843
\(533\) −6141.83 −0.499122
\(534\) 863.184 0.0699506
\(535\) −16193.4 −1.30860
\(536\) 2454.38 0.197786
\(537\) −6799.14 −0.546377
\(538\) 758.260 0.0607637
\(539\) −2728.62 −0.218052
\(540\) 849.924 0.0677313
\(541\) 18834.0 1.49674 0.748372 0.663280i \(-0.230836\pi\)
0.748372 + 0.663280i \(0.230836\pi\)
\(542\) −17873.2 −1.41646
\(543\) −1347.27 −0.106477
\(544\) 13859.5 1.09232
\(545\) −12392.8 −0.974037
\(546\) −1432.92 −0.112314
\(547\) −16024.5 −1.25258 −0.626289 0.779591i \(-0.715427\pi\)
−0.626289 + 0.779591i \(0.715427\pi\)
\(548\) 3024.03 0.235730
\(549\) 2160.27 0.167938
\(550\) −1380.43 −0.107021
\(551\) 14423.7 1.11519
\(552\) 7271.31 0.560666
\(553\) 8867.29 0.681872
\(554\) −15725.0 −1.20594
\(555\) 6116.47 0.467802
\(556\) −8000.47 −0.610244
\(557\) 12161.8 0.925157 0.462578 0.886578i \(-0.346924\pi\)
0.462578 + 0.886578i \(0.346924\pi\)
\(558\) 1034.32 0.0784697
\(559\) 4752.49 0.359587
\(560\) −3651.92 −0.275575
\(561\) −5785.99 −0.435445
\(562\) 3842.64 0.288420
\(563\) 20165.9 1.50958 0.754790 0.655967i \(-0.227739\pi\)
0.754790 + 0.655967i \(0.227739\pi\)
\(564\) −6218.26 −0.464248
\(565\) 10405.9 0.774831
\(566\) 11515.0 0.855144
\(567\) 1146.89 0.0849466
\(568\) −6467.57 −0.477770
\(569\) −18351.0 −1.35205 −0.676024 0.736880i \(-0.736298\pi\)
−0.676024 + 0.736880i \(0.736298\pi\)
\(570\) −4639.46 −0.340922
\(571\) −19894.1 −1.45804 −0.729021 0.684492i \(-0.760024\pi\)
−0.729021 + 0.684492i \(0.760024\pi\)
\(572\) −977.304 −0.0714391
\(573\) −1485.62 −0.108312
\(574\) −12153.4 −0.883748
\(575\) −3284.62 −0.238223
\(576\) 4626.80 0.334693
\(577\) 916.177 0.0661022 0.0330511 0.999454i \(-0.489478\pi\)
0.0330511 + 0.999454i \(0.489478\pi\)
\(578\) 11366.5 0.817968
\(579\) 6354.56 0.456108
\(580\) −6107.51 −0.437242
\(581\) −100.319 −0.00716340
\(582\) −9619.18 −0.685099
\(583\) −2092.16 −0.148625
\(584\) 7273.16 0.515352
\(585\) 1339.67 0.0946813
\(586\) 300.298 0.0211693
\(587\) −1121.91 −0.0788862 −0.0394431 0.999222i \(-0.512558\pi\)
−0.0394431 + 0.999222i \(0.512558\pi\)
\(588\) −1404.78 −0.0985240
\(589\) 3934.81 0.275265
\(590\) −1227.36 −0.0856432
\(591\) −1116.81 −0.0777314
\(592\) 5728.70 0.397717
\(593\) 16327.9 1.13070 0.565352 0.824850i \(-0.308740\pi\)
0.565352 + 0.824850i \(0.308740\pi\)
\(594\) −1122.39 −0.0775292
\(595\) 13665.6 0.941573
\(596\) −4075.00 −0.280065
\(597\) −2969.23 −0.203555
\(598\) 3336.71 0.228174
\(599\) 24940.5 1.70124 0.850620 0.525780i \(-0.176227\pi\)
0.850620 + 0.525780i \(0.176227\pi\)
\(600\) −2441.13 −0.166098
\(601\) 17458.9 1.18496 0.592482 0.805584i \(-0.298148\pi\)
0.592482 + 0.805584i \(0.298148\pi\)
\(602\) 9404.17 0.636687
\(603\) −901.460 −0.0608794
\(604\) −11119.2 −0.749064
\(605\) −9240.25 −0.620941
\(606\) 8791.99 0.589357
\(607\) 19831.7 1.32610 0.663052 0.748573i \(-0.269261\pi\)
0.663052 + 0.748573i \(0.269261\pi\)
\(608\) 10227.9 0.682230
\(609\) −8241.47 −0.548377
\(610\) 4993.26 0.331428
\(611\) −9801.37 −0.648971
\(612\) −2978.82 −0.196751
\(613\) 4099.73 0.270125 0.135062 0.990837i \(-0.456877\pi\)
0.135062 + 0.990837i \(0.456877\pi\)
\(614\) 5351.26 0.351725
\(615\) 11362.5 0.745006
\(616\) −6642.64 −0.434480
\(617\) 19906.4 1.29887 0.649434 0.760418i \(-0.275006\pi\)
0.649434 + 0.760418i \(0.275006\pi\)
\(618\) −8195.39 −0.533442
\(619\) −11172.3 −0.725450 −0.362725 0.931896i \(-0.618154\pi\)
−0.362725 + 0.931896i \(0.618154\pi\)
\(620\) −1666.14 −0.107926
\(621\) −2670.65 −0.172576
\(622\) −20139.0 −1.29823
\(623\) −1876.31 −0.120662
\(624\) 1254.74 0.0804964
\(625\) −10371.4 −0.663769
\(626\) 12374.3 0.790060
\(627\) −4269.88 −0.271966
\(628\) −642.905 −0.0408514
\(629\) −21437.0 −1.35890
\(630\) 2650.92 0.167643
\(631\) 23430.1 1.47819 0.739096 0.673600i \(-0.235253\pi\)
0.739096 + 0.673600i \(0.235253\pi\)
\(632\) −15345.9 −0.965867
\(633\) 3804.85 0.238909
\(634\) −1867.21 −0.116966
\(635\) −6532.97 −0.408273
\(636\) −1077.11 −0.0671545
\(637\) −2214.25 −0.137726
\(638\) 8065.46 0.500493
\(639\) 2375.45 0.147060
\(640\) 149.254 0.00921844
\(641\) −16450.0 −1.01363 −0.506816 0.862055i \(-0.669177\pi\)
−0.506816 + 0.862055i \(0.669177\pi\)
\(642\) 11009.5 0.676808
\(643\) −11207.2 −0.687355 −0.343677 0.939088i \(-0.611673\pi\)
−0.343677 + 0.939088i \(0.611673\pi\)
\(644\) −4601.52 −0.281561
\(645\) −8792.17 −0.536731
\(646\) 16260.4 0.990335
\(647\) 28416.4 1.72668 0.863341 0.504621i \(-0.168368\pi\)
0.863341 + 0.504621i \(0.168368\pi\)
\(648\) −1984.83 −0.120326
\(649\) −1129.59 −0.0683207
\(650\) −1120.20 −0.0675969
\(651\) −2248.30 −0.135357
\(652\) −446.812 −0.0268382
\(653\) −16811.3 −1.00747 −0.503734 0.863859i \(-0.668041\pi\)
−0.503734 + 0.863859i \(0.668041\pi\)
\(654\) 8425.61 0.503773
\(655\) −13162.4 −0.785184
\(656\) 10642.1 0.633391
\(657\) −2671.33 −0.158628
\(658\) −19394.8 −1.14907
\(659\) −7979.14 −0.471659 −0.235829 0.971794i \(-0.575781\pi\)
−0.235829 + 0.971794i \(0.575781\pi\)
\(660\) 1808.03 0.106632
\(661\) 19673.4 1.15765 0.578824 0.815453i \(-0.303512\pi\)
0.578824 + 0.815453i \(0.303512\pi\)
\(662\) 22344.7 1.31186
\(663\) −4695.28 −0.275037
\(664\) 173.614 0.0101469
\(665\) 10084.8 0.588078
\(666\) −4158.46 −0.241947
\(667\) 19191.2 1.11407
\(668\) 2061.89 0.119426
\(669\) −3389.33 −0.195873
\(670\) −2083.64 −0.120146
\(671\) 4595.50 0.264393
\(672\) −5844.07 −0.335476
\(673\) −12964.9 −0.742588 −0.371294 0.928515i \(-0.621086\pi\)
−0.371294 + 0.928515i \(0.621086\pi\)
\(674\) 4041.44 0.230965
\(675\) 896.593 0.0511257
\(676\) 6425.34 0.365575
\(677\) 26269.4 1.49131 0.745655 0.666332i \(-0.232137\pi\)
0.745655 + 0.666332i \(0.232137\pi\)
\(678\) −7074.74 −0.400743
\(679\) 20909.2 1.18177
\(680\) −23650.0 −1.33373
\(681\) 483.629 0.0272139
\(682\) 2200.28 0.123538
\(683\) −8043.98 −0.450650 −0.225325 0.974284i \(-0.572344\pi\)
−0.225325 + 0.974284i \(0.572344\pi\)
\(684\) −2198.27 −0.122885
\(685\) −8818.18 −0.491862
\(686\) −14926.5 −0.830751
\(687\) −356.941 −0.0198226
\(688\) −8234.77 −0.456319
\(689\) −1697.77 −0.0938751
\(690\) −6172.96 −0.340580
\(691\) 3316.01 0.182557 0.0912785 0.995825i \(-0.470905\pi\)
0.0912785 + 0.995825i \(0.470905\pi\)
\(692\) 9427.72 0.517902
\(693\) 2439.75 0.133735
\(694\) −5086.90 −0.278236
\(695\) 23329.6 1.27330
\(696\) 14262.9 0.776772
\(697\) −39823.2 −2.16415
\(698\) −20642.3 −1.11937
\(699\) 13408.5 0.725547
\(700\) 1544.82 0.0834127
\(701\) −13786.8 −0.742824 −0.371412 0.928468i \(-0.621126\pi\)
−0.371412 + 0.928468i \(0.621126\pi\)
\(702\) −910.813 −0.0489692
\(703\) −15819.9 −0.848731
\(704\) 9842.50 0.526922
\(705\) 18132.7 0.968674
\(706\) 15994.8 0.852652
\(707\) −19111.2 −1.01662
\(708\) −581.547 −0.0308699
\(709\) −5117.29 −0.271063 −0.135532 0.990773i \(-0.543274\pi\)
−0.135532 + 0.990773i \(0.543274\pi\)
\(710\) 5490.63 0.290225
\(711\) 5636.34 0.297299
\(712\) 3247.18 0.170917
\(713\) 5235.40 0.274989
\(714\) −9290.96 −0.486983
\(715\) 2849.85 0.149061
\(716\) −7446.36 −0.388664
\(717\) 14048.0 0.731704
\(718\) 16662.7 0.866079
\(719\) −5501.98 −0.285382 −0.142691 0.989767i \(-0.545575\pi\)
−0.142691 + 0.989767i \(0.545575\pi\)
\(720\) −2321.28 −0.120151
\(721\) 17814.3 0.920167
\(722\) −2893.09 −0.149127
\(723\) 11509.8 0.592055
\(724\) −1475.52 −0.0757420
\(725\) −6442.87 −0.330044
\(726\) 6282.24 0.321151
\(727\) −23570.5 −1.20245 −0.601226 0.799079i \(-0.705321\pi\)
−0.601226 + 0.799079i \(0.705321\pi\)
\(728\) −5390.45 −0.274428
\(729\) 729.000 0.0370370
\(730\) −6174.53 −0.313054
\(731\) 30814.8 1.55914
\(732\) 2365.91 0.119463
\(733\) 30142.7 1.51889 0.759446 0.650571i \(-0.225470\pi\)
0.759446 + 0.650571i \(0.225470\pi\)
\(734\) 10521.6 0.529101
\(735\) 4096.39 0.205575
\(736\) 13608.6 0.681546
\(737\) −1917.66 −0.0958450
\(738\) −7725.08 −0.385317
\(739\) −17267.8 −0.859548 −0.429774 0.902937i \(-0.641407\pi\)
−0.429774 + 0.902937i \(0.641407\pi\)
\(740\) 6698.72 0.332770
\(741\) −3464.97 −0.171780
\(742\) −3359.53 −0.166216
\(743\) 36299.4 1.79232 0.896161 0.443728i \(-0.146344\pi\)
0.896161 + 0.443728i \(0.146344\pi\)
\(744\) 3890.95 0.191733
\(745\) 11882.8 0.584368
\(746\) −7985.12 −0.391898
\(747\) −63.7661 −0.00312327
\(748\) −6336.78 −0.309753
\(749\) −23931.4 −1.16747
\(750\) 9873.38 0.480700
\(751\) −19115.5 −0.928807 −0.464404 0.885624i \(-0.653731\pi\)
−0.464404 + 0.885624i \(0.653731\pi\)
\(752\) 16983.1 0.823550
\(753\) 993.364 0.0480746
\(754\) 6545.05 0.316123
\(755\) 32424.1 1.56296
\(756\) 1256.06 0.0604266
\(757\) 7724.96 0.370896 0.185448 0.982654i \(-0.440626\pi\)
0.185448 + 0.982654i \(0.440626\pi\)
\(758\) 5316.38 0.254749
\(759\) −5681.22 −0.271693
\(760\) −17453.0 −0.833009
\(761\) 15791.5 0.752223 0.376112 0.926574i \(-0.377261\pi\)
0.376112 + 0.926574i \(0.377261\pi\)
\(762\) 4441.63 0.211159
\(763\) −18314.8 −0.868990
\(764\) −1627.04 −0.0770475
\(765\) 8686.33 0.410529
\(766\) 16887.2 0.796552
\(767\) −916.649 −0.0431529
\(768\) 12236.7 0.574938
\(769\) −29233.6 −1.37086 −0.685430 0.728139i \(-0.740386\pi\)
−0.685430 + 0.728139i \(0.740386\pi\)
\(770\) 5639.25 0.263928
\(771\) −4271.45 −0.199523
\(772\) 6959.46 0.324451
\(773\) −37733.7 −1.75574 −0.877869 0.478901i \(-0.841035\pi\)
−0.877869 + 0.478901i \(0.841035\pi\)
\(774\) 5977.61 0.277598
\(775\) −1757.63 −0.0814658
\(776\) −36186.0 −1.67397
\(777\) 9039.25 0.417350
\(778\) −28571.4 −1.31663
\(779\) −29388.3 −1.35166
\(780\) 1467.20 0.0673514
\(781\) 5053.25 0.231523
\(782\) 21635.0 0.989343
\(783\) −5238.56 −0.239094
\(784\) 3836.69 0.174776
\(785\) 1874.73 0.0852383
\(786\) 8948.80 0.406098
\(787\) −13387.8 −0.606384 −0.303192 0.952929i \(-0.598052\pi\)
−0.303192 + 0.952929i \(0.598052\pi\)
\(788\) −1223.12 −0.0552941
\(789\) 15616.2 0.704627
\(790\) 13027.9 0.586723
\(791\) 15378.4 0.691268
\(792\) −4222.29 −0.189435
\(793\) 3729.21 0.166996
\(794\) −16276.8 −0.727510
\(795\) 3140.90 0.140121
\(796\) −3251.88 −0.144799
\(797\) 38966.4 1.73182 0.865911 0.500199i \(-0.166740\pi\)
0.865911 + 0.500199i \(0.166740\pi\)
\(798\) −6856.44 −0.304155
\(799\) −63551.4 −2.81388
\(800\) −4568.67 −0.201909
\(801\) −1192.64 −0.0526092
\(802\) −9437.99 −0.415545
\(803\) −5682.67 −0.249735
\(804\) −987.272 −0.0433064
\(805\) 13418.2 0.587489
\(806\) 1785.51 0.0780295
\(807\) −1047.67 −0.0456998
\(808\) 33074.2 1.44003
\(809\) −22024.2 −0.957145 −0.478572 0.878048i \(-0.658846\pi\)
−0.478572 + 0.878048i \(0.658846\pi\)
\(810\) 1685.01 0.0730931
\(811\) 4588.64 0.198679 0.0993397 0.995054i \(-0.468327\pi\)
0.0993397 + 0.995054i \(0.468327\pi\)
\(812\) −9026.00 −0.390087
\(813\) 24695.0 1.06530
\(814\) −8846.20 −0.380908
\(815\) 1302.92 0.0559991
\(816\) 8135.64 0.349025
\(817\) 22740.4 0.973789
\(818\) 31778.1 1.35831
\(819\) 1979.84 0.0844702
\(820\) 12444.1 0.529958
\(821\) −45052.8 −1.91517 −0.957584 0.288154i \(-0.906959\pi\)
−0.957584 + 0.288154i \(0.906959\pi\)
\(822\) 5995.29 0.254391
\(823\) −17701.8 −0.749752 −0.374876 0.927075i \(-0.622315\pi\)
−0.374876 + 0.927075i \(0.622315\pi\)
\(824\) −30829.9 −1.30341
\(825\) 1907.30 0.0804895
\(826\) −1813.85 −0.0764068
\(827\) 10882.0 0.457562 0.228781 0.973478i \(-0.426526\pi\)
0.228781 + 0.973478i \(0.426526\pi\)
\(828\) −2924.88 −0.122761
\(829\) 26577.8 1.11349 0.556746 0.830683i \(-0.312050\pi\)
0.556746 + 0.830683i \(0.312050\pi\)
\(830\) −147.389 −0.00616381
\(831\) 21726.9 0.906975
\(832\) 7987.10 0.332816
\(833\) −14357.0 −0.597169
\(834\) −15861.3 −0.658552
\(835\) −6012.53 −0.249188
\(836\) −4676.34 −0.193463
\(837\) −1429.09 −0.0590163
\(838\) −13661.8 −0.563171
\(839\) −13400.2 −0.551402 −0.275701 0.961243i \(-0.588910\pi\)
−0.275701 + 0.961243i \(0.588910\pi\)
\(840\) 9972.39 0.409619
\(841\) 13255.0 0.543483
\(842\) −23415.7 −0.958383
\(843\) −5309.29 −0.216918
\(844\) 4167.05 0.169947
\(845\) −18736.5 −0.762788
\(846\) −12328.0 −0.500999
\(847\) −13655.7 −0.553975
\(848\) 2941.77 0.119128
\(849\) −15910.0 −0.643146
\(850\) −7263.32 −0.293094
\(851\) −21048.9 −0.847880
\(852\) 2601.57 0.104611
\(853\) −3838.21 −0.154065 −0.0770327 0.997029i \(-0.524545\pi\)
−0.0770327 + 0.997029i \(0.524545\pi\)
\(854\) 7379.31 0.295685
\(855\) 6410.24 0.256404
\(856\) 41416.2 1.65371
\(857\) 44265.5 1.76439 0.882194 0.470886i \(-0.156066\pi\)
0.882194 + 0.470886i \(0.156066\pi\)
\(858\) −1937.55 −0.0770944
\(859\) −26682.8 −1.05984 −0.529921 0.848047i \(-0.677779\pi\)
−0.529921 + 0.848047i \(0.677779\pi\)
\(860\) −9629.12 −0.381803
\(861\) 16792.0 0.664659
\(862\) 12928.6 0.510845
\(863\) −3259.02 −0.128550 −0.0642748 0.997932i \(-0.520473\pi\)
−0.0642748 + 0.997932i \(0.520473\pi\)
\(864\) −3714.69 −0.146269
\(865\) −27491.6 −1.08063
\(866\) 9904.84 0.388661
\(867\) −15704.9 −0.615186
\(868\) −2462.32 −0.0962862
\(869\) 11990.1 0.468050
\(870\) −12108.4 −0.471855
\(871\) −1556.16 −0.0605379
\(872\) 31695.9 1.23092
\(873\) 13290.6 0.515257
\(874\) 15966.0 0.617914
\(875\) −21461.8 −0.829190
\(876\) −2925.62 −0.112840
\(877\) −37049.0 −1.42652 −0.713259 0.700900i \(-0.752782\pi\)
−0.713259 + 0.700900i \(0.752782\pi\)
\(878\) −15852.7 −0.609341
\(879\) −414.916 −0.0159212
\(880\) −4938.02 −0.189160
\(881\) 7446.48 0.284765 0.142383 0.989812i \(-0.454524\pi\)
0.142383 + 0.989812i \(0.454524\pi\)
\(882\) −2785.04 −0.106323
\(883\) 23801.2 0.907106 0.453553 0.891229i \(-0.350156\pi\)
0.453553 + 0.891229i \(0.350156\pi\)
\(884\) −5142.24 −0.195647
\(885\) 1695.81 0.0644114
\(886\) −34334.6 −1.30191
\(887\) 20503.7 0.776154 0.388077 0.921627i \(-0.373140\pi\)
0.388077 + 0.921627i \(0.373140\pi\)
\(888\) −15643.5 −0.591174
\(889\) −9654.77 −0.364242
\(890\) −2756.68 −0.103825
\(891\) 1550.79 0.0583090
\(892\) −3711.97 −0.139334
\(893\) −46899.0 −1.75746
\(894\) −8078.89 −0.302235
\(895\) 21713.9 0.810966
\(896\) 220.576 0.00822426
\(897\) −4610.26 −0.171608
\(898\) 3484.79 0.129498
\(899\) 10269.4 0.380982
\(900\) 981.942 0.0363682
\(901\) −11008.2 −0.407034
\(902\) −16433.4 −0.606622
\(903\) −12993.5 −0.478846
\(904\) −26614.2 −0.979176
\(905\) 4302.66 0.158039
\(906\) −22044.4 −0.808362
\(907\) −10620.2 −0.388797 −0.194399 0.980923i \(-0.562276\pi\)
−0.194399 + 0.980923i \(0.562276\pi\)
\(908\) 529.666 0.0193586
\(909\) −12147.7 −0.443250
\(910\) 4576.20 0.166703
\(911\) −6121.02 −0.222611 −0.111305 0.993786i \(-0.535503\pi\)
−0.111305 + 0.993786i \(0.535503\pi\)
\(912\) 6003.85 0.217990
\(913\) −135.648 −0.00491709
\(914\) 21527.2 0.779057
\(915\) −6899.09 −0.249264
\(916\) −390.919 −0.0141008
\(917\) −19452.0 −0.700504
\(918\) −5905.65 −0.212326
\(919\) −50071.6 −1.79729 −0.898645 0.438677i \(-0.855447\pi\)
−0.898645 + 0.438677i \(0.855447\pi\)
\(920\) −23221.8 −0.832174
\(921\) −7393.72 −0.264529
\(922\) −12413.2 −0.443393
\(923\) 4100.66 0.146235
\(924\) 2672.00 0.0951323
\(925\) 7066.54 0.251185
\(926\) 34373.8 1.21986
\(927\) 11323.4 0.401196
\(928\) 26693.6 0.944245
\(929\) 39484.7 1.39446 0.697228 0.716849i \(-0.254416\pi\)
0.697228 + 0.716849i \(0.254416\pi\)
\(930\) −3303.21 −0.116469
\(931\) −10595.0 −0.372974
\(932\) 14684.9 0.516117
\(933\) 27825.6 0.976387
\(934\) 814.637 0.0285393
\(935\) 18478.3 0.646314
\(936\) −3426.35 −0.119651
\(937\) −50370.5 −1.75617 −0.878085 0.478504i \(-0.841179\pi\)
−0.878085 + 0.478504i \(0.841179\pi\)
\(938\) −3079.31 −0.107189
\(939\) −17097.3 −0.594197
\(940\) 19858.8 0.689065
\(941\) −18333.3 −0.635119 −0.317560 0.948238i \(-0.602863\pi\)
−0.317560 + 0.948238i \(0.602863\pi\)
\(942\) −1274.59 −0.0440853
\(943\) −39102.1 −1.35031
\(944\) 1588.30 0.0547615
\(945\) −3662.72 −0.126083
\(946\) 12716.0 0.437034
\(947\) −12078.7 −0.414473 −0.207236 0.978291i \(-0.566447\pi\)
−0.207236 + 0.978291i \(0.566447\pi\)
\(948\) 6172.88 0.211483
\(949\) −4611.44 −0.157738
\(950\) −5360.10 −0.183058
\(951\) 2579.88 0.0879688
\(952\) −34951.3 −1.18989
\(953\) 40916.7 1.39079 0.695395 0.718628i \(-0.255230\pi\)
0.695395 + 0.718628i \(0.255230\pi\)
\(954\) −2135.43 −0.0724707
\(955\) 4744.51 0.160763
\(956\) 15385.3 0.520497
\(957\) −11143.9 −0.376416
\(958\) −7820.03 −0.263730
\(959\) −13032.0 −0.438816
\(960\) −14776.2 −0.496772
\(961\) −26989.5 −0.905961
\(962\) −7178.62 −0.240590
\(963\) −15211.6 −0.509021
\(964\) 12605.5 0.421157
\(965\) −20294.0 −0.676983
\(966\) −9122.73 −0.303850
\(967\) 27358.4 0.909812 0.454906 0.890539i \(-0.349673\pi\)
0.454906 + 0.890539i \(0.349673\pi\)
\(968\) 23632.9 0.784701
\(969\) −22466.6 −0.744822
\(970\) 30720.0 1.01687
\(971\) −25783.7 −0.852151 −0.426076 0.904688i \(-0.640104\pi\)
−0.426076 + 0.904688i \(0.640104\pi\)
\(972\) 798.395 0.0263462
\(973\) 34477.8 1.13598
\(974\) −30684.9 −1.00945
\(975\) 1547.76 0.0508390
\(976\) −6461.70 −0.211920
\(977\) 17329.6 0.567475 0.283738 0.958902i \(-0.408425\pi\)
0.283738 + 0.958902i \(0.408425\pi\)
\(978\) −885.826 −0.0289628
\(979\) −2537.09 −0.0828249
\(980\) 4486.33 0.146235
\(981\) −11641.5 −0.378883
\(982\) −636.231 −0.0206751
\(983\) −23593.4 −0.765527 −0.382764 0.923846i \(-0.625028\pi\)
−0.382764 + 0.923846i \(0.625028\pi\)
\(984\) −29060.7 −0.941484
\(985\) 3566.65 0.115374
\(986\) 42437.7 1.37068
\(987\) 26797.4 0.864206
\(988\) −3794.81 −0.122195
\(989\) 30256.9 0.972813
\(990\) 3584.50 0.115074
\(991\) 27698.5 0.887863 0.443931 0.896061i \(-0.353583\pi\)
0.443931 + 0.896061i \(0.353583\pi\)
\(992\) 7282.07 0.233071
\(993\) −30873.2 −0.986639
\(994\) 8114.34 0.258925
\(995\) 9482.60 0.302129
\(996\) −69.8362 −0.00222173
\(997\) 21554.5 0.684691 0.342346 0.939574i \(-0.388779\pi\)
0.342346 + 0.939574i \(0.388779\pi\)
\(998\) 33809.3 1.07236
\(999\) 5745.65 0.181966
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 177.4.a.c.1.6 8
3.2 odd 2 531.4.a.f.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.c.1.6 8 1.1 even 1 trivial
531.4.a.f.1.3 8 3.2 odd 2